INTERNATIONAL GONGRESS X∙(A)=Hom(G_(m),A_( bar(F)))\mathbb{X} \bullet(A)=\operatorname{Hom}\left(\mathbb{G}_{m}, A_{\bar{F}}\right) the group of its cocharacters.
For a prime â„“\ell, let Lambda\Lambda be F_(â„“),Z_(â„“),Q_(â„“)\mathbb{F}_{\ell}, \mathbb{Z}_{\ell}, \mathbb{Q}_{\ell} or a finite (flat) extension of such rings. It will serve as the coefficient ring of our sheaf theory.
1. FROM CLASSICAL TO GEOMETRIC LANGLANDS CORRESPONDENCE
In this section, we review some developments of the geometric Langlands theory inspired from the classical theory, with another important source of inspiration from quantum physics. The basic idea is categorification/geometrization, which is a process of replacing set-theoretic statements with categorical analogues
We illustrate this process by some important examples.
1.1. The geometric Satake
The starting point of the Langlands program is (Langlands' interpretation of) the Satake isomorphism, in which the Langlands dual group appears mysteriously. Similarly, the starting point of the geometric Langlands theory is the geometric Satake equivalence, which is a tensor equivalence between the category of perverse sheaves on the (spherical) local Hecke stack of a connected reductive group and the category of finite-dimensional algebraic representations of its dual group. This is a vast generalization of the classical Satake isomorphism (via the sheaf-to-function dictionary), and arguably gives a conceptual explanation why the Langlands dual group (in fact, the CC-group) should appear in the Langlands correspondence.
We follow [83, SECT. 1.1] for notations and conventions regarding dual groups. Let GG denote a connected reductive group over a field FF. Let ( hat(G), hat(B), hat(T), hat(e))(\hat{G}, \hat{B}, \hat{T}, \hat{e}) be a pinned Langlands dual group of GG over Z\mathbb{Z}. There is a finite Galois extension tilde(F)//F\tilde{F} / F, and a natural injective map xi:Gamma_( tilde(F)//F)sub Aut( hat(G), hat(B), hat(T), hat(e))\xi: \Gamma_{\tilde{F} / F} \subset \operatorname{Aut}(\hat{G}, \hat{B}, \hat{T}, \hat{e}), induced by the action of Gamma_(F)\Gamma_{F} on the root datum of GG. Let ^(L)G= hat(G)><|Gamma_( tilde(F)//F){ }^{L} G=\hat{G} \rtimes \Gamma_{\tilde{F} / F} denote the usual LL-group of GG, and ^(c)G= hat(G)><|(G_(m)xxGamma_( tilde(F)//F)){ }^{c} G=\hat{G} \rtimes\left(\mathbb{G}_{m} \times \Gamma_{\tilde{F} / F}\right) the group defined in [83], which is isomorphic to the CC-group of GG introduced by Buzzard-Gee. We write d:^(c)G rarrG_(m)xxGamma_( tilde(F)//F)d:{ }^{c} G \rightarrow \mathbb{G}_{m} \times \Gamma_{\tilde{F} / F} for the projection with the kernel hat(G)\hat{G}.
Now let FF be a nonarchimedean local field with O\mathcal{O} being its ring of integers and k=F_(q)k=\mathbb{F}_{q} its residue field. That is, FF is a finite extension of Q_(p)\mathbb{Q}_{p} or is isomorphic to F_(q)((Ï–))\mathbb{F}_{q}((\varpi)). Let sigma\sigma be the geometric qq-Frobenius of kk. Assume that GG can be extended to a connected reductive group over O\mathcal{O} (such GG is called unramified), and we fix such an extension to have G(O)sub G(F)G(\mathcal{O}) \subset G(F), usually called a hyperspecial subgroup of G(F)G(F). With a basis of open neighborhoods of the unit given by finite-index subgroups of G(O)G(\mathcal{O}), the group G(F)G(F) is a locally compact topological group. The classical spherical Hecke algebra is the space of compactly supported G(O)G(\mathcal{O})-biinvariant C\mathbb{C}-valued functions on G(F)G(F), equipped with the convolution product
{:(1.2)(f**g)(x)=int_(G(F))f(y)g(y^(-1)x)dy:}\begin{equation*}
(f * g)(x)=\int_{G(F)} f(y) g\left(y^{-1} x\right) d y \tag{1.2}
\end{equation*}
where dyd y is the Haar measure on G(F)G(F) such that G(O)G(\mathcal{O}) has volume 1 . Note that if both ff and gg are Z\mathbb{Z}-valued, so is f**gf * g. Therefore, the subset H_(G(O))^(cl)H_{G(\mathcal{O})}^{\mathrm{cl}} of Z\mathbb{Z}-valued functions forms a Z\mathbb{Z}-algebra. ^(1){ }^{1}
On the dual side, under the unramifiedness assumption, Gamma_( tilde(F)//F)\Gamma_{\tilde{F} / F} is a finite cyclic group generated by sigma\sigma. Note that hat(G)\hat{G} acts on ^(c)G|_(d=(q,sigma))\left.{ }^{c} G\right|_{d=(q, \sigma)}, the fiber of dd at (q,sigma)inG_(m)xxGamma_( tilde(F)//F)(q, \sigma) \in \mathbb{G}_{m} \times \Gamma_{\tilde{F} / F}, by conjugation. Then the classical Satake isomorphism establishes a canonical isomorphism of Z[q^(-1)]\mathbb{Z}\left[q^{-1}\right]-algebras
Remark 1.1.1. In fact, as explained in [83], there is a Satake isomorphism over Z\mathbb{Z} (without inverting qq ), in which the CC-group ^(c)G{ }^{c} G is replaced by certain affine monoid containing it as the group of invertible elements. On the other hand, if we extend the base ring from Z[q^(-1)]\mathbb{Z}\left[q^{-1}\right] to Z[q^(+-(1)/(2))]\mathbb{Z}\left[q^{ \pm \frac{1}{2}}\right], one can rewrite (1.3) as an isomorphism
where hat(G)\hat{G} acts on hat(G)sigma sub^(L)G\hat{G} \sigma \subset{ }^{L} G by the usual conjugation (e.g., see [83] for the discussion). This is the more traditional formulation of the Satake isomorphism, which is slightly noncanonical, but suffices for many applications.
In the geometric theory, where instead of thinking G(F)G(F) as a topological group and considering the space of G(O)G(\mathcal{O})-biinvariant compactly supported functions on it, one regards
1 Here (-)^(cl)(-)^{\mathrm{cl}} stands for the classical Hecke algebra, as opposed to the derived Hecke algebra mentioned in (2.2). G(F)G(F) as a certain algebro-geometric object and studies the category of G(O)G(\mathcal{O})-biequivariant sheaves on it. In the rest of the section, we allow FF to be slightly more general. Namely, we assume that FF is a local field complete with respect to a discrete valuation, with ring of integers O\mathcal{O} and a perfect residue field kk of characteristic p > 0.^(2)p>0 .{ }^{2} Let Ï–inO\varpi \in \mathcal{O} be a uniformizer.
For a perfect kk-algebra RR, let W_(O)(R)W_{\mathcal{O}}(R) denote the ring of Witt vectors in RR with coefficient in O\mathcal{O}. If char F=F= char kk, then W_(O)(R)≃R[[ϖ]]W_{\mathcal{O}}(R) \simeq R[[\varpi]]. If char F!=F \neq char kk, see [78, SECT. 0.5]. If R= bar(k)R=\bar{k}, we denote W_(O)( bar(k))W_{\mathcal{O}}(\bar{k}) by O_(F^(˘))\mathcal{O}_{\breve{F}} and W_(O)( bar(k))[1//ϖ]W_{\mathcal{O}}(\bar{k})[1 / \varpi] by F^(˘)\breve{F}. We write D_(R)=Spec W_(O)(R)D_{R}=\operatorname{Spec} W_{\mathcal{O}}(R) and D_(R)^(**)=Spec W_(O)(R)[1//ϖ]D_{R}^{*}=\operatorname{Spec} W_{\mathcal{O}}(R)[1 / \varpi] which are thought as a family of (punctured) discs parameterized by Spec R\operatorname{Spec} R.
We denote by L^(+)GL^{+} G (resp. LGL G ) the jet group (resp. loop group) of GG. As presheaves on Aff _(k)^("pf "){ }_{k}^{\text {pf }},
L^(+)G(R)=G(W_(O)(R)),quad LG(R)=G(W_(O)(R)[1//Ï–])L^{+} G(R)=G\left(W_{\mathcal{O}}(R)\right), \quad L G(R)=G\left(W_{\mathcal{O}}(R)[1 / \varpi]\right)
Note that L^(+)G(k)=G(O)L^{+} G(k)=G(\mathcal{O}) and LG(k)=G(F)L G(k)=G(F). Let
Hk_(G):=L^(+)G\\LG//L^(+)G\mathrm{Hk}_{G}:=L^{+} G \backslash L G / L^{+} G
The space Gr_(G)\operatorname{Gr}_{G} is usually called the affine Grassmannian of GG. See [4,23] for the equal characteristic case and [13,78][13,78] for the mixed characteristic case, and see [77,80] for examples of closed subvarieties in Gr_(G)\mathrm{Gr}_{G}. The theorem allows one to define the category of constructible and perverse sheaves on Hk_(G)\mathrm{Hk}_{G}, and to formulate the geometric Satake, as we discuss now.
First, the (left) quotient by L^(+)GL^{+} G-action induces a map Gr_(G)rarrHk_(G)\mathrm{Gr}_{G} \rightarrow \mathrm{Hk}_{G}. Roughly speaking, a sheaf on Hk_(G)\mathrm{Hk}_{G} is perverse (resp. constructible) if its pullback to Gr_(G)\mathrm{Gr}_{G} comes from a perverse (resp. constructible) sheaf on some X_(i)X_{i}. Then inside Shv(Hk_(G),Lambda)\mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right) we have the categories Perv(Hk_(G),Lambda)subShv_(c)(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right) \subset \operatorname{Shv}_{c}\left(\mathrm{Hk}_{G}, \Lambda\right) of perverse and constructible sheaves on Hk_(G)\mathrm{Hk}_{G}. They can be regarded as categorical analogues of the space of G(O)G(\mathcal{O})-biinvariant compactly supported functions on G(F)G(F). In addition, Perv(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right) is an abelian category, semisimple if Lambda\Lambda is a field of characteristic zero, called the Satake category of GG. For simplicity, we assume that Lambda\Lambda is a field in the sequel. ^(3){ }^{3}
There is also a categorical analogue of the convolution product (1.2). Namely, there is the convolution diagram
Hk_(G)xxHk_(G)larr^(pr)L^(+)G\\LG xx^(L^(+)G)LG//L^(+)Grarr"m"Hk_(G)\mathrm{Hk}_{G} \times \mathrm{Hk}_{G} \stackrel{\mathrm{pr}}{\leftarrow} L^{+} G \backslash L G \times{ }^{L^{+} G} L G / L^{+} G \xrightarrow{m} \mathrm{Hk}_{G}
and the convolution of two sheaves A,BinShv(Hk_(G),Lambda)\mathscr{A}, \mathscr{B} \in \mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right) is defined as
This convolution product makes Shv(Hk_(G),Lambda)\mathbf{S h v}\left(\mathrm{Hk}_{G}, \Lambda\right) into a monoidal oo\infty-category containing Perv(Hk_(G),Lambda)subShv_(c)(Hk_(G),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G}, \Lambda\right) \subset \operatorname{Shv}_{c}\left(\mathrm{Hk}_{G}, \Lambda\right) as monoidal subcategories.
Remark 1.1.3. The above construction of the Satake category as a monoidal category is essentially equivalent to the more traditional approach, in which the Satake category is defined as the category of L^(+)GL^{+} G-equivariant perverse sheaves on Gr_(G)\operatorname{Gr}_{G} (e.g., see [80] for an exposition).
Let Coh (B hat(G)_(Lambda))^(rho)\operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}\right)^{\rho} denote the abelian monoidal category of coherent sheaves on the classifying stack B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda} over Lambda^(4)\Lambda^{4}, which is equivalent to the category of algebraic representations of hat(G)\hat{G} on finite dimensional Lambda\Lambda-vector spaces. This following theorem is usually known as the geometric Satake equivalence.
Theorem 1.1.4. There is a canonical equivalence of monoidal abelian categories
3 The formulation for Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell} is slightly more complicated, as the right-hand side of (1.5) may not be perverse when A\mathscr{A} and B\mathscr{B} are perverse.
4 In the dual group side, we always work in the realm of usual algebraic geometry, so B hat(G)\mathbb{B} \hat{G} is an Artin stack in the usual sense.
Geometric Satake is really one of the cornerstones of the geometric Langlands program, and has found numerous applications to representation theory, mathematical physics, and (arithmetic) algebraic geometry. When F=k((ϖ))F=k((\varpi)), this theorem grew out of works of Lusztig, Ginzburg, Beilinson-Drinfeld and Mirković-Vilonen (cf. [5,51,53]). In mixed characteristic, it was proved in [69,78][69,78], with the equal characteristic case as an input, and in [25] by mimicking the strategy in equal characteristic. We conclude this subsection with a few remarks.
Remark 1.1.5. (1) As mentioned before, the geometric Satake can be regarded as the conceptual definition of the Langlands dual group hat(G)\hat{G} of GG, namely as the Tannakian group of the Tannakian category Perv(Hk_(G)ox( bar(k)),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right). In addition, as explained in [72, 76], the group hat(G)\hat{G} is canonically equipped with a pinning ( hat(B), hat(T), hat(e))(\hat{B}, \hat{T}, \hat{e}). In the rest of the article, by the pinned Langlands dual group ( hat(G), hat(B), hat(T), hat(e))\hat{G}, \hat{B}, \hat{T}, \hat{e}) of GG, we mean the quadruple defined by the geometric Satake.
(2) For arithmetic applications, one needs to understand the Gamma_(k)\Gamma_{k}-action on Perv(Hk_(G)ox( bar(k)),Lambda)\operatorname{Perv}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right) in terms of the dual group side. It turns out that such an action is induced by an action of Gamma_(k)\Gamma_{k} on hat(G)\hat{G}, preserving ( hat(B), hat(T))(\hat{B}, \hat{T}) but not hat(e)\hat{e}. See [76, 80], or [77] from the motivic point of view. This leads to the appearance of the group ^(c)G{ }^{c} G. See [76,80,83][76,80,83] for detailed discussions.
(3) There is also the derived Satake equivalence [11], describing Shv_(c)(Hk_(G)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{G} \otimes \bar{k}, \Lambda\right) in terms of the dual group, at least when Lambda\Lambda is a field of characteristic zero. We mention that the category in the dual side is not the derived category of coherent sheaves on B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}.
(4) In fact, for many applications, it is important to have a family version of the geometric Satake. For a (nonempty) finite set SS, there is a local Hecke stack Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}} over D^(S)D^{S}, the self-product of the disc D=Spec OD=\operatorname{Spec} \mathcal{O} over SS, which, roughly speaking, classifies quadruples ({x_(s)}_(s in S),E,E^('),beta)\left(\left\{x_{s}\right\}_{s \in S}, \mathcal{E}, \mathcal{E}^{\prime}, \beta\right), where {x_(s)}_(s in S)\left\{x_{s}\right\}_{s \in S} is an SS-tuple of points of D,ED, \mathcal{E} and E^(')\mathcal{E}^{\prime} are two GG-torsors on DD, and beta\beta is an isomorphism between E\mathcal{E} and E^(')\mathcal{E}^{\prime} on D-uuu_(s){x_(s)}D-\bigcup_{s}\left\{x_{s}\right\}. In equal characteristic, one can regard DD as the formal disc at a kk-point of an algebraic curve XX over kk and Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}} is the restriction along D^(S)rarrX^(S)D^{S} \rightarrow X^{S} of the stack
where (LG)_(X^(S))(L G)_{X^{S}} and (L^(+)G)_(X^(S))\left(L^{+} G\right)_{X^{S}} are family versions of LGL G and L^(+)GL^{+} G over X^(S)X^{S} (e.g., see [80, SECT. 3.1] for precise definitions). In mixed characteristic, the stack Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}} (and in fact D^(S)D^{S} itself) does not live in the world of (perfect) algebraic geometry, but rather in the world of perfectoid analytic geometry as developed by Scholze (see [25,59]). In both cases, one can consider certain category Perv ^(ULA)(Hk_(G,D^(S))ox( bar(k)),Lambda)^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S}} \otimes \bar{k}, \Lambda\right) of (ULA) perverse sheaves on Hk_(G,D^(s))ox bar(k)\mathrm{Hk}_{G, D^{s}} \otimes \bar{k}. In addition, for a map S rarrS^(')S \rightarrow S^{\prime} of finite sets, restriction along Hk_(G,D^(S^(')))rarrHk_(G,D^(s))\mathrm{Hk}_{G, D^{S^{\prime}}} \rightarrow \mathrm{Hk}_{G, D^{s}} gives a functor Perv ^(ULA)(Hk_(G,D^(S))ox( bar(k)),Lambda)rarrPerv^(ULA)(Hk_(G,D^(S^(')))ox( bar(k)),Lambda).^(5){ }^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S}} \otimes \bar{k}, \Lambda\right) \rightarrow \operatorname{Perv}^{\mathrm{ULA}}\left(\mathrm{Hk}_{G, D^{S^{\prime}}} \otimes \bar{k}, \Lambda\right) .{ }^{5} We refer to the above mentioned references for details.
On the other hand, let hat(G)^(S)\hat{G}^{S} be the SS-power self-product of hat(G)\hat{G} over Lambda\Lambda. Then for S rarrS^(')S \rightarrow S^{\prime}, the restriction along B hat(G)^(S^('))rarrB hat(G)^(S)\mathbb{B} \hat{G}^{S^{\prime}} \rightarrow \mathbb{B} \hat{G}^{S} gives a functor Coh (B hat(G)_(Lambda)^(S))^(â—¯)rarr Coh (B hat(G)_(Lambda)^(S^(')))^(â—¯)\operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}^{S}\right)^{\bigcirc} \rightarrow \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\Lambda}^{S^{\prime}}\right)^{\bigcirc}. Now a family version of the geometric Satake gives a system of functors
compatible with restriction functors on both sides induced by maps between finite sets (see [28,80][28,80] ).
(5) For applications, it is important to have the geometric Satake in different sheaftheoretic contents over different versions of local Hecke stacks. Besides the above mentioned ones, we also mention a DD-module version [5], and an arithmetic DD-module version [66].
1.2. Tamely ramified local geometric Langlands correspondence
We first recall the classical theory. Assume that FF is a finite extension of Q_(p)\mathbb{Q}_{p} or is isomorphic to F_(q)((Ï–))\mathbb{F}_{q}((\varpi)), and for simplicity assume that GG extends to a connected reductive group over O\mathcal{O}. (In fact, results in the subsection hold in appropriate forms for quasi-split groups that are split over a tamely ramified extension of FF.) In addition, we fix a pinning (B,T,e)(B, T, e) of GG over O\mathcal{O}.
The classical local Langlands program aims to classify (smooth) irreducible representations of G(F)G(F) (over C\mathbb{C} ) in terms of Galois representations. From this point of view, the Satake isomorphism (1.3) gives a classification of irreducible unramified representations of G(F)G(F), i.e., those that have nonzero vectors fixed by G(O)G(\mathcal{O}), as such representations are in one-to-one correspondence with simple modules of H_(G(O))^(cl)oxCH_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{C}, which via the Satake isomorphism (1.3) are parameterized by semisimple hat(G)\hat{G}-conjugacy classes in ^(c)G{ }^{c} G. (For an irreducible unramified representation pi\pi, the corresponding hat(G)\hat{G}-conjugacy class in ^(c)G{ }^{c} G is usually called the Satake parameter of pi\pi.)
The next important class of irreducible representations are those that have nonzero vectors fixed by an Iwahori subgroup G(F)G(F). For example, under the reduction mod Ï–\varpi map G(O)rarr G(k)G(\mathcal{O}) \rightarrow G(k), the preimage II of B(k)sub G(k)B(k) \subset G(k) is an Iwahori subgroup of G(F)G(F). As in the unramified case, the Z\mathbb{Z}-valued II-biinvariant functions form a Z\mathbb{Z}-algebra H_(I)^(cl)H_{I}^{\mathrm{cl}} with multiplication given by convolution (1.2) (with the Haar measure dgd g chosen so that the volume of II is one), and the set of irreducible representations of G(F)G(F) that have nonzero II-fixed vectors are in bijection with the set of simple (H_(I)^(cl)oxC)\left(H_{I}^{\mathrm{cl}} \otimes \mathbb{C}\right)-modules. Just as the Satake isomorphism, Kazhdan-Lusztig gave a description of H_(I)^(cl)oxCH_{I}^{\mathrm{cl}} \otimes \mathbb{C} in terms of geometric objects associated to hat(G)\hat{G}.
Let hat(U)sub hat(B)\hat{U} \subset \hat{B} denote the unipotent radical of hat(B)\hat{B}. The natural morphism hat(U)// hat(B)rarr hat(G)// hat(G)\hat{U} / \hat{B} \rightarrow \hat{G} / \hat{G} is usually called the Springer resolution. Let
and therefore on S_( hat(G),C)^("unip ")S_{\hat{G}, \mathbb{C}}^{\text {unip }}, by identifying hat(U)_(C)\hat{U}_{\mathbb{C}} with its Lie algebra via the exponential map. Then
one can form the quotient stack S_( hat(G),C)^("unip ")//G_(m,C)S_{\hat{G}, \mathbb{C}}^{\text {unip }} / \mathbb{G}_{m, \mathbb{C}}. In the sequel, for an Artin stack XX (of finite presentation) over C\mathbb{C}, we let K(X)K(X) denote the KK-group of the (oo(\infty-)category of coherent sheaves on XX.
Kazhdan-Lusztig [36] constructed (under the assumption that GG is split with connected center) a canonical isomorphism (after choosing a square root of sqrtq\sqrt{q} of qq )
where the map K(BG_(m,C))rarrCK\left(\mathbb{B} \mathbb{G}_{m, \mathbb{C}}\right) \rightarrow \mathbb{C} sends the class corresponding to the tautological representation of G_(m,C)\mathbb{G}_{m, \mathbb{C}} to sqrtq\sqrt{q}. In addition, the isomorphism induces the Bernstein isomorphism
where Z(H_(I)^(cl)oxC)Z\left(H_{I}^{\mathrm{cl}} \otimes \mathbb{C}\right) is the center of H_(I)^(cl)oxCH_{I}^{\mathrm{cl}} \otimes \mathbb{C}, and the map K(B hat(G)_(C))rarr K(S_( hat(G),C)^("unip ")//G_(m,C))K\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right) \rightarrow K\left(S_{\hat{G}, \mathbb{C}}^{\text {unip }} / \mathbb{G}_{m, \mathbb{C}}\right) is induced by the natural projection S_( hat(G))^("unip ")//G_(m)rarrB hat(G)S_{\hat{G}}^{\text {unip }} / \mathbb{G}_{m} \rightarrow \mathbb{B} \hat{G}.
Remark 1.2.1. It would be interesting to give a description of the Z\mathbb{Z}-algebra H_(I)^(cl)H_{I}^{\mathrm{cl}} in terms of the geometry involving hat(G)\hat{G}, which would generalize the integral Satake isomorphism from [83].
It turns out that the Kazhdan-Lusztig isomorphism (1.7) also admits a categorification, usually known as the Bezrukavnikov equivalence, which gives two realizations of the affine Hecke category. Again, when switching to the geometric theory, we allow FF to be a little bit more general as in Section 1.1. We also assume that GG extends to a connected reductive group over O\mathcal{O} and fix a pinning of GG over O\mathcal{O}. Let L^(+)G rarrG_(k)L^{+} G \rightarrow G_{k} be the natural "reduction mod Ï–\varpi " map, and let Iw subL^(+)G\subset L^{+} G be the preimage of B_(k)subG_(k)B_{k} \subset G_{k}. This is the geometric analogue of II. Then as in the unramified case discussed in Section 1.1, one can define the Iwahori local Hecke stack Hk_(Iw)=Iw\\LG//Iw\mathrm{Hk}_{\mathrm{Iw}}=\mathrm{Iw} \backslash L G / \mathrm{Iw} and the monoidal categories Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)subShv(Hk_(Iw)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right) \subset \mathbf{S h v}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right). The category Shvv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\mathbf{S h v} v_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right) can be thought as a categorical analogue of H_(I)^(cl)H_{I}^{\mathrm{cl}}, usually called the affine Hecke category.
Recall that we let F^(˘)=W_(O)( bar(k))[1//ϖ]\breve{F}=W_{\mathcal{O}}(\bar{k})[1 / \varpi]. The inertia I_(F):=Gamma_(F^(˘))I_{F}:=\Gamma_{\breve{F}} of FF has a tame quotient I_(F)^(t)I_{F}^{t} isomorphic to prod_(ℓ!=p)Z_(ℓ)(1)\prod_{\ell \neq p} \mathbb{Z}_{\ell}(1).
Theorem 1.2.2. For every choice of a topological generator tau\tau of the tame inertia I_(F)^(t)I_{F}^{t}, there is a canonical equivalence of monoidal oo\infty-categories
In fact, Bezrukavnikov proved such equivalence when F=k((Ï–))F=k((\varpi)) in [9]. Yun and the author deduced the mixed characteristic case from the equal characteristic case. It would be interesting to know whether the new techniques introduced in [25,59] can lead a direct proof of this equivalence in mixed characteristic. (See [1] for some progress in this direction.)
Remark 1.2.3. Again, for arithmetic applications, one needs to describe the action of Gamma_(k)\Gamma_{k} on Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\mathbf{S h} \mathbf{v}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right) in terms of the dual group side. See [9,35][9,35] for a discussion.
We explain an important ingredient in the proof of Theorem 1.2.2 (when F=k((Ï–)))F=k((\varpi))). There is a smooth affine group scheme E\mathscr{E} (called the Iwahori group scheme)
over DD, analogous to Hk_(G,D^(S))\mathrm{Hk}_{G, D^{S}} as discussed at the end of Section 1.1 (here S={1}S=\{1\} ). In addition, Hk_(G,D)|_(D^(**))~=Hk_(G,D)|_(D^(**))\left.\left.\mathrm{Hk}_{\mathscr{G}, D}\right|_{D^{*}} \cong \mathrm{Hk}_{G, D}\right|_{D^{*}} and Hk_(G,D)|_(0)=Hk_(Iw)\left.\mathrm{Hk}_{\mathscr{G}, D}\right|_{0}=\mathrm{Hk}_{\mathrm{Iw}}, where 0in D0 \in D is the closed point. Then taking nearby cycles gives
This is known as Gaitsgory's central functor [27,75], which can be regarded as a categorification of (1.8). We remark this construction is motivated by the Kottwitz conjecture originated from the study of mod pp geometry of Shimura varieties. See Section 3.1 for some discussions.
Theorem 1.2.2 admits a generalization to the tame level. We consider the following diagram:
where the left morphism is the usual Grothendieck-Springer resolution. Let chi\chi be a Lambda\Lambda-point of hat(T)// hat(T)\hat{T} / \hat{T}, where Lambda\Lambda is a finite extension of Q_(â„“)\mathbb{Q}_{\ell}. Let ( hat(B)// hat(B))_(chi)=q_( hat(B))^(-1)(chi)(\hat{B} / \hat{B})_{\chi}=q_{\hat{B}}^{-1}(\chi), and let
Note that if chi=1\chi=1, this reduces to S_( hat(G),Lambda)^("unip ")S_{\hat{G}, \Lambda}^{\text {unip }}. On the other hand, a (torsion) Lambda\Lambda-point chi in hat(T)// hat(T)\chi \in \hat{T} / \hat{T} defines a one-dimensional character sheaf L_(chi)\mathscr{L}_{\chi} on Iw ox bar(k)\otimes \bar{k}. Then one can define the monoidal category of bi-(Iw, L_(chi)\mathscr{L}_{\chi} )-equivariant constructible sheaves on LG_( bar(k))L G_{\bar{k}}, denoted as Sh_("cons ")(chi(Hk_(Iw))_(chi),Lambda)\operatorname{Sh}_{\text {cons }}\left(\chi\left(\mathrm{Hk}_{\mathrm{Iw}}\right)_{\chi}, \Lambda\right). If chi=1\chi=1, so L_(chi)\mathscr{L}_{\chi} is the trivial character sheaf on Iw, this reduces to the affine Hecke category Shv_(c)(Hk_(Iw)ox( bar(k)),Lambda)\operatorname{Shv}_{c}\left(\mathrm{Hk}_{\mathrm{Iw}} \otimes \bar{k}, \Lambda\right). The following generalization of Theorem 1.2.2 is conjectured in [9] and will be proved in a forthcoming joint work with Dhillon-Li-Yun [18].
Theorem 1.2.4. Assume that char F=char kF=\operatorname{char} k. There is a canonical monoidal equivalence
Remark 1.2.5. It is important to establish a version of equivalences in Theorems 1.2.2 and 1.2.4 for Z_(â„“)\mathbb{Z}_{\ell}-sheaves.
Remark 1.2.6. The local geometric Langlands correspondence beyond the tame ramification has not been fully understood, although certain wild ramifications have appeared in concrete problems (e.g., [31,79]). It is widely believed that the general local geometric Langlands should be formulated as 2-categorical statement, predicting the 2-category of module categories under the action of (appropriately defined) category of sheaves on LGL G is equivalent to the 2-category of categories over the stack of local geometric Langlands parameters. The precise formulation is beyond the scope of this survey, but, roughly speaking, it implies (and is more or less equivalent to saying) that the Hecke category for appropriately chosen "level" of LGL G is (Morita) equivalent to the category of coherent sheaves on some stack of the form Xxx_(Y)XX \times_{Y} X, where YY is closely related to the moduli of local geometric Langlands parameters.
1.3. Global geometric Langlands correspondence
As mentioned at the beginning of the article, the (global) geometric Langlands program originated from Drinfeld's proof of Langlands conjecture for GL_(2)\mathrm{GL}_{2} over function fields. Early developments of this subject mostly focused on the construction of Hecke eigensheaves associated to Galois representations of a global function field FF (or, equivalently, local systems on a smooth algebraic curve XX ), e.g., see [20,26,42][20,26,42].
The scope of the whole program then shifted after the work [5], in which BeilinsonDrinfeld formulated a rough categorical form of the global geometric Langlands correspondence. The formulation then was made precise by Arinkin-Gaitsgory in [2], which we now recall. Let XX be a smooth projective curve over F=CF=\mathbb{C}. On the automorphic side, let D_(c)(Bun_(G))\mathbf{D}_{c}\left(\mathrm{Bun}_{G}\right) be the oo\infty-category of coherent D-modules on the moduli stack Bun _(G){ }_{G} of principal GG-bundles on XX. On the Galois side, let Coh(Loc_( hat(G)))\operatorname{Coh}\left(\operatorname{Loc}_{\hat{G}}\right) be the oo\infty-category of coherent sheaves on the moduli stack Loc_( hat(G))\operatorname{Loc}_{\hat{G}} of de Rham hat(G)\hat{G}-local systems (also known as principal hat(G)\hat{G}-bundles with flat connection) on XX.
Conjecture 1.3.1. There is a canonical equivalence of oo\infty-categories
satisfying a list of natural compatibility conditions.
We briefly mention the most important compatibility condition, and refer to [2] for the rest. Note that both sides admit actions by a family of commuting operators labeled by x in Xx \in X and V in Coh (B hat(G)_(C))^(oo)V \in \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right)^{\infty}. Namely, for every point x in Xx \in X, there is the evaluation map Loc_( hat(G))rarrB hat(G)_(C)\operatorname{Loc}_{\hat{G}} \rightarrow \mathbb{B} \hat{G}_{\mathbb{C}} so every V in Coh (B hat(G)_(C))^(o.)V \in \operatorname{Coh}\left(\mathbb{B} \hat{G}_{\mathbb{C}}\right)^{\odot} gives a vector bundle tilde(V)_(x)\tilde{V}_{x} on Loc_( hat(G))\operatorname{Loc}_{\hat{G}} by pullback, which then acts on Coh(Loc_( hat(G)))\operatorname{Coh}\left(\operatorname{Loc}_{\hat{G}}\right) by tensoring. On the other hand, there is the Hecke operator H_(V,x)H_{V, x} that acts on D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right) by convolving the sheaf Sat_({1})(V)|_(x)\left.\operatorname{Sat}_{\{1\}}(V)\right|_{x} from the ( DD-module version of) the geometric Satake (1.6). Then the equivalence L_(G)\mathbb{L}_{G} should intertwine the actions of these operators.
Although the conjecture remains widely open, it is known that the category of perfect complexes Perf(Loc_( hat(G)))\operatorname{Perf}\left(\operatorname{Loc}_{\hat{G}}\right) on Loc_( hat(G))\operatorname{Loc}_{\hat{G}} acts on D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right), usually called the spectral action, in a way such that the action of tilde(V)_(x)in Perf(Loc_( hat(G)))\tilde{V}_{x} \in \operatorname{Perf}\left(\operatorname{Loc}_{\hat{G}}\right) on D_(c)(Bun_(G))\mathbf{D}_{c}\left(\operatorname{Bun}_{G}\right) is given by the Hecke operator H_(V,x)H_{V, x}.
Nowadays, Conjecture 1.3.1 sometimes is referred as the de Rham version of the global geometric Langlands conjecture, as there are some other versions of such conjectural equivalences, which we briefly mention.
First, in spirit of the nonabelian Hodge theory, there should exist a classical limit of Conjecture 1.3.1, sometimes known as the Dolbeault version of the global geometric Langlands. While the precise formulation is unknown (to the author), generically, it amounts to the duality of Hitchin fibrations for GG and hat(G)\hat{G} (in the sense of mirror symmetry), and was established "generically" in [15,19]. By twisting/deforming such duality in positive characteristic, one can prove a characteristic pp analogue of Conjecture 1.3.1 (of course, only "generically," see [10,14,15])[10,14,15]). Interestingly, this "generic" characteristic pp equivalence can be used to give a new proof of the main result of [5] (at least for G=GL_(n)G=\mathrm{GL}_{n}, see [12]).
The work [5] (and therefore the de Rham version of the global geometric Langlands) was strongly influenced by conformal field theory. On the other hand, motivated by topological field theory, Ben-Zvi and Nadler [7] proposed a Betti version of Conjecture 1.3.1, where on the automorphic side the category of DD-modules on Bun_(G)\mathrm{Bun}_{G} is replaced with the category of sheaves of C\mathbb{C}-vector spaces on (the analytification of) Bun _(G){ }_{G} and on the Galois side Loc_( hat(G))\operatorname{Loc}_{\hat{G}} is replaced by the moduli of Betti hat(G)\hat{G}-local systems (also known as hat(G)\hat{G}-valued representations of fundamental group of XX ).
The Riemann-Hilbert correspondence allows passing between the de Rham hat(G)\hat{G}-local systems and Betti hat(G)\hat{G}-local systems, but in a transcendental way. So Conjecture 1.3.1 and its Betti analogue are not directly related. Recently, Arinkin et al. [3] proposed yet another version of Conjecture 1.3.1, which directly relates both de Rham and Betti versions, and at the same time includes a version in terms of â„“\ell-adic sheaves. So it is more closely related to the classical Langlands correspondence over function fields, as will be discussed in Section 2.2.
2. FROM GEOMETRIC TO CLASSICAL LANGLANDS PROGRAM
In the previous section, we discussed how the ideas of categorification and geometrization led to the developments of the geometric Langlands program. On the other hand, the ideas of quantum physics allow one to reverse arrows in (1.1) by evaluating a (topological) quantum field theory at manifolds of different dimensions. Such ideas are certainly not new in geometry and topology. But, surprisingly, they also lead to a new understanding of the classical Langlands program. Indeed, it has been widely known that there is an analogy between global fields and 3-manifolds, and under such analogy Frobenius corresponds to the fundamental group of a circle. Then "compactification of field theories on a circle" leads to the categorical trace method (e.g., see [3,6,77][3,6,77] ), which plays a more and more important role in the geometric representation theory.
2.1. Categorical arithmetic local Langlands
In this subsection, let FF be either a finite extension of Q_(p)\mathbb{Q}_{p} or isomorphic to F_(q)((Ï–))\mathbb{F}_{q}((\varpi)). The classical local Langlands correspondence seeks a classification of smooth irreducible representations of G(F)G(F) in terms of Galois data. The precise formulation, beyond the G=GL_(n)G=\mathrm{GL}_{n} case (which is a theorem by [30,43][30,43] ), is complicated. However, the yoga that the local geometric Langlands is 2-categorical (see Remark 1.2.6) suggests that the even the classical local Langlands correspondence should and probably needs to be categorified.
The first ingredient needed to formulate the categorical arithmetic local Langlands is the following result, due independently to [17,25,82][17,25,82]. We take the formulation from [82] and refer for the notion of (strongly) continuous homomorphisms to the same reference.
Theorem 2.1.1. The prestack sending a Z_(â„“)\mathbb{Z}_{\ell}-algebra AA to the space of (strongly) continuous homomorphisms rho:W_(F)rarr^(c)G(A)\rho: W_{F} \rightarrow{ }^{c} G(A) such that d@rho=(cycl^(-1),pr)d \circ \rho=\left(\mathrm{cycl}^{-1}, \mathrm{pr}\right) is represented by a (classical) scheme Loc_(c)^(â—»)\operatorname{Loc}_{c}^{\square}, which is a disjoint union of affine schemes that are flat, of finite type, and of locally complete intersection over Z_(â„“)\mathbb{Z}_{\ell}.
The conjugation action of hat(G)\hat{G} on ^(c)G{ }^{c} G induces an action of hat(G)\hat{G} on Loc_(c)^(â—»)\operatorname{Loc}_{c}^{\square}, and we call the quotient stack Loc_(_(G))=Loc_(_(c))^(â—»)// hat(G)\operatorname{Loc}_{{ }_{G}}=\operatorname{Loc}_{{ }_{c}}^{\square} / \hat{G} the stack of local Langlands parameters, which, roughly speaking, classifies the groupoid of the above rho\rho 's up to hat(G)\hat{G}-conjugacy.
In the categorical version of the local Langlands correspondence, on the Galois side it is natural to consider the (oo:}\left(\infty\right.-)category Coh(Loc_(c))\operatorname{Coh}\left(\operatorname{Loc}_{c}\right) of coherent sheaves on Loc_(c)_(G)\operatorname{Loc}_{c}{ }_{G}. On the representation side, one might naively consider the (oo(\infty-)category Rep(G(F),Lambda)\boldsymbol{\operatorname { R e p }}(G(F), \Lambda) of smooth representations of G(F)G(F). But in fact, this category needs to be enlarged. This can be seen from different point of view. Indeed, it is a general wisdom shared by many people that in the classical local Langlands correspondence, it is better to study representations of GG together with a collection of groups that are (refined version of) its inner forms. There are various proposals of such collections. Arithmetic geometry (i.e., the study of pp-adic and mod pp geometry of Shimura varieties and moduli of Shtukas) and geometric representation theory (i.e., the categorical trace construction) suggest studying a category glued from the categories of representations of a collection of groups {J_(b)(F)}_(b in B(G))\left\{J_{b}(F)\right\}_{b \in B(G)} arising from the Kottwitz set
B(G)=G(F^(˘))//∼,quad" where "g∼g^(')" if "g^(')=h^(-1)g sigma(h)" for some "h in G(F^(˘))B(G)=G(\breve{F}) / \sim, \quad \text { where } g \sim g^{\prime} \text { if } g^{\prime}=h^{-1} g \sigma(h) \text { for some } h \in G(\breve{F})
Here for b in B(G)b \in B(G) (lifted to an element in G(F^(˘))G(\breve{F}) ), the group J_(b)J_{b} is an FF-group defined by assigning and FF-algebra the group J_(b)(R)={h in G((F^(˘))ox_(F)R)∣h^(-1)b sigma(h)=b}J_{b}(R)=\left\{h \in G\left(\breve{F} \otimes_{F} R\right) \mid h^{-1} b \sigma(h)=b\right\}. In particular, if b=1b=1 then J_(b)=GJ_{b}=G. In general, there is a well-defined embedding (J_(b))_( bar(F))rarrG_( bar(F))\left(J_{b}\right)_{\bar{F}} \rightarrow G_{\bar{F}} up to conjugacy, making J_(b)J_{b} a refinement of an inner form of a Levi subgroup of GG (say, GG is quasisplit).
B(G):=LG//Ad_(sigma)LG\mathfrak{B}(G):=L G / \operatorname{Ad}_{\sigma} L G
where Ad_(sigma)\operatorname{Ad}_{\sigma} denotes the Frobenius twisted conjugation given by
Ad_(sigma):LG xx LG rarr LG,quad(h,g)|->hg sigma(h)^(-1)\operatorname{Ad}_{\sigma}: L G \times L G \rightarrow L G, \quad(h, g) \mapsto h g \sigma(h)^{-1}
Then we have the category of Lambda\Lambda-sheaves Shv(B(G)ox bar(k),Lambda)\boldsymbol{\operatorname { S h v }}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda) as mentioned before. Although B(G)\mathfrak{B}(G) is a wild object in the traditional algebraic geometry, there are still a few things one can say about its geometry, and the category Shv(B(G)ox bar(k),Lambda)\operatorname{Shv}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda) is quite reasonable. In addition, it is possible to define the category Shv_(c)(B(G)ox bar(k),Lambda)\mathbf{S h v}_{c}(\mathfrak{B}(G) \otimes \bar{k}, \Lambda) of constructible sheaves on B(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k}, as we now briefly explain and refer to [35] for careful discussions.
For every algebraically closed field Omega\Omega over kk, the groupoid of Omega\Omega-points of B(G)\mathfrak{B}(G) is the groupoid of FF-isocrystals with GG-structure over Omega\Omega and the set of its isomorphism classes can be identified with the Kottwitz set B(G)B(G). However, B(G)\mathfrak{B}(G) is not merely a disjoint union
of its points. Rather, it admits a stratification, known as the Newton stratification, labeled by B(G)B(G). Namely, the set B(G)B(G) has a natural partial order and, roughly speaking, for each b in B(G)b \in B(G) those Omega\Omega-points corresponding to b^(') <= bb^{\prime} \leq b form a closed substack i_( <= b):B(G)_( <= b)subi_{\leq b}: \mathfrak{B}(G)_{\leq b} \subsetB(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k} and those Omega\Omega-points corresponding to bb form an open substack j_(b):B(G)_(b)subj_{b}: \mathfrak{B}(G)_{b} \subsetB(G)_( <= b)\mathfrak{B}(G)_{\leq b}. In particular, basic elements in B(G)B(G) (i.e., minimal elements with respect to the partial order <=\leq ) give closed strata. We also mention that if bb is basic, J_(b)J_{b} is a refinement of an inner form of GG, usually called an extended pure inner form of GG.
In the rest of this subsection, we simply denote B(G)ox bar(k)\mathfrak{B}(G) \otimes \bar{k} by B(G)\mathfrak{B}(G). We write i_(b)=i_( <= b)j_(b):B(G)_(b)↪B(G)i_{b}=i_{\leq b} j_{b}: \mathfrak{B}(G)_{b} \hookrightarrow \mathfrak{B}(G) for the locally closed embedding. For bb, let Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right) be the full subcategory of Rep(J_(b)(F),Lambda)\operatorname{Rep}\left(J_{b}(F), \Lambda\right) generated (under finite colimits and retracts) by compactly induced representations
from the trivial representation of open compact subgroups K subJ_(b)(F)K \subset J_{b}(F). The following theorem from [35] summarizes some properties of Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda).
Theorem 2.1.2. (1) An object in Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda) is constructible if and only if its !-restriction to each B(G)_(b)\mathfrak{B}(G)_{b} is constructible and is zero for almost all b's. If Lambda\Lambda is a field of characteristic zero, Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) consist of compact objects in Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda).
(2) For every b in B(G)b \in B(G), there is a canonical equivalence Shv_(c)(B(G)_(b),Lambda)~=\operatorname{Shv}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right) \congRep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right). There are fully faithful embeddings i_(b,**),i_(b,!)i_{b, *}, i_{b,!} : Shv_(c)(B(G)_(b),Lambda)rarrShv_(c)(B(G),Lambda)\operatorname{Shv}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right) \rightarrow \operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) (which coincide when bb is basic), inducing a semiorthogonal decomposition of Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) in terms of {Shv_(c)(B(G)_(b),Lambda)}_(b)\left\{\mathbf{S h v}_{c}\left(\mathfrak{B}(G)_{b}, \Lambda\right)\right\}_{b}.
(3) There is a self-duality functor D^("coh "):Shv_(c)(B(G),Lambda)≃Shv_(c)(B(G),Lambda)^(vv)ob-\mathbb{D}^{\text {coh }}: \operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) \simeq \mathbf{S h v}_{c}(\mathfrak{B}(G), \Lambda)^{\vee} o b- tained by gluing cohomological dualities (in the sense of Bernstein-Zelevinsky) on various Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right) 's.
(4) There is a natural perverse tt-structure obtained by gluing (shifted) tt-structures on various Rep_("f.g. ")(J_(b)(F),Lambda)\operatorname{Rep}_{\text {f.g. }}\left(J_{b}(F), \Lambda\right) 's, preserved by D^("coh ")\mathbb{D}^{\text {coh }} if Lambda\Lambda is a field.
The following categorical form of the arithmetic local Langlands conjecture [82, SEcT. 4.6] is inspired by the global geometric Langlands conjecture as discussed in Section 1.3 .
Conjecture 2.1.3. Assume that GG is quasisplit over FF equipped with a pinning (B,T,e)(B, T, e) and fix a nontrivial additive character psi:F rarrZ_(â„“)[mu_(p^(oo))]^(xx)\psi: F \rightarrow \mathbb{Z}_{\ell}\left[\mu_{p^{\infty}}\right]^{\times}. There is a canonical equivalence of categories
Remark 2.1.4. (1) There is a closely related version of the conjecture, with Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) replaced by Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda) and with Coh(Loc_(G)ox Lambda:}\operatorname{Coh}\left(\operatorname{Loc}_{G} \otimes \Lambda\right. ) replaced by its ind-completion (with
certain support condition imposed) (see [82, SECT. 4.6]). Fargues-Scholze [25] make a conjecture parallel to this version, with the category Shv(B(G),Lambda)\operatorname{Shv}(\mathfrak{B}(G), \Lambda) replaced by D_("lis ")(Bun_(G),Lambda)D_{\text {lis }}\left(\operatorname{Bun}_{G}, \Lambda\right) as mentioned above.
(2) It is also explained in [82] a motivic hope to have a version of such equivalence over Q\mathbb{Q}.
One consequence of the conjecture is that for every bb there should exist a fully faithful embedding
obtained as the restriction of a quasinverse of L_(G)\mathbb{L}_{G} to i_(b,!)(Rep_(f.g.)(J_(b)(F),Lambda))i_{b,!}\left(\operatorname{Rep}_{\mathrm{f} . \mathrm{g} .}\left(J_{b}(F), \Lambda\right)\right). The existence of such functor is closely related to the idea of local Langlands in families and has also been considered (in the case J_(b)=GJ_{b}=G is split and Lambda\Lambda is a field of characteristic zero) in [6,32].
In particular, for every open compact subgroup K subJ_(b)(F)K \subset J_{b}(F) there should exist a coherent sheaf
on Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \Lambda such that
The algebra H_(K,Lambda)H_{K, \Lambda} is sometimes called the derived Hecke algebra as it might not concentrate on cohomological degree zero (when Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell} or F_(â„“)\mathbb{F}_{\ell} ). See [82, SECTS. 4.3-4.5] for conjectural descriptions of A_(K,Lambda)\mathfrak{A}_{K, \Lambda} in various cases.
As in the global geometric Langlands conjecture, the equivalence from Conjecture 2.1.3 should satisfy a set of compatibility conditions. For example, it should be compatible with parabolic inductions on both sides, and should be compatible with the duality D^("coh ")\mathbb{D}^{\text {coh }} on Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) and the (modified) Grothendieck-Serre duality of Coh(Loc_(c)ox Lambda)\operatorname{Coh}\left(\operatorname{Loc}_{c} \otimes \Lambda\right). We refer to [35,82][35,82] for more details.
On the other hand, Conjecture 2.1.3 predicts an action of the category Perf(Loc_(G)ox Lambda)\operatorname{Perf}\left(\operatorname{Loc}_{G} \otimes \Lambda\right) of perfect complexes on Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \Lambda on Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda), analogous to the spectral action as mentioned in Section 1.3. One of the main results of [25] is the construction of such action in their setting. Currently the existence of such a spectral action on Shv_(c)(B(G),Lambda)\operatorname{Shv}_{c}(\mathfrak{B}(G), \Lambda) is not known. But there are convincing evidences that Conjecture 2.1.3 should still be true.
We assume that GG extends to a reductive group over O\mathcal{O} as before. Then there are closed substacks
usually called the stack of unramified parameters (resp. unipotent parameters), classifying those rho\rho such that rho(I_(F))\rho\left(I_{F}\right) is trivial (resp. rho(I_(F))\rho\left(I_{F}\right) is unipotent). For Lambda=Q_(â„“),Loc_(c)^("unip ")oxQ_(â„“)\Lambda=\mathbb{Q}_{\ell}, \operatorname{Loc}_{c}^{\text {unip }} \otimes \mathbb{Q}_{\ell} is a connected component of Loc^(G)_(G)oxQ_(â„“)\operatorname{Loc}^{G}{ }_{G} \otimes \mathbb{Q}_{\ell}.
On the other hand, there is the unipotent subcategory Shv_(c)^("unip ")(B(G),Q_(â„“))sub\operatorname{Shv}_{c}^{\text {unip }}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right) \subsetSh_(c)(B(G),Q_(â„“))\operatorname{Sh}_{c}\left(\mathfrak{B}(G), \mathbb{Q}_{\ell}\right), which roughly speaking is the glue of categories Rep_("f.g. ")^("unip ")(J_(b)(F),Q_(â„“))\operatorname{Rep}_{\text {f.g. }}^{\text {unip }}\left(J_{b}(F), \mathbb{Q}_{\ell}\right)
of unipotent representations of J_(b)(F)J_{b}(F) (introduced in [52]) for all b in B(G)b \in B(G). We have the following theorem from [35], deduced from Theorem 1.2.2 by taking the Frobenius-twisted categorical trace.
Theorem 2.1.5. For a reductive group GG over O\mathcal{O} with a fixed pinning (B,T,e)(B, T, e), there is a canonical equivalence
For arithmetic applications, it is important to match specific objects under the equivalence. We give a few examples and refer to [35] for many more of such matchings (see also [82, SECTS. 4.3-4.5]).
Example 2.1.6. The equivalence L_(G)^("unip ")\mathbb{L}_{G}^{\text {unip }} gives the the conjectural coherent sheaf in (2.1) for all parahoric subgroups K sub G(F)K \subset G(F) (in the sense of Bruhat-Tits) such that (2.2) holds. For example, we have V_(G(O),Q_(â„“))~=O_("Loc "_(C_(G))^(ur))oxQ_(â„“)\mathfrak{V}_{G(\mathcal{O}), \mathbb{Q}_{\ell}} \cong \mathcal{O}_{\text {Loc }_{C_{G}}^{\mathrm{ur}}} \otimes \mathbb{Q}_{\ell}, which gives
As Loc_(cG)^(ur)~=(^(c)G|_(d=(q,sigma)))// hat(G)\operatorname{Loc}_{c G}^{\mathrm{ur}} \cong\left(\left.{ }^{c} G\right|_{d=(q, \sigma)}\right) / \hat{G}, taking the 0 th cohomology recovers the Satake isomorphism (1.3). In addition, it implies that the left-hand side has no higher cohomology, which is not obvious. We mention that it is conjectured in [82, SECT. 4.3] that V_(G(O),Z_(â„“))~=O_("Loc "_(c_(G)))\mathfrak{V}_{G(\mathcal{O}), \mathbb{Z}_{\ell}} \cong \mathcal{O}_{\text {Loc }_{c_{G}}} so the first isomorphism in (2.3) should hold over Z_(â„“)\mathbb{Z}_{\ell}, known as the (conjectural) derived Satake isomorphism. (But H_(G(O),Z_(â„“))!=H_(G(O))^(cl)oxZ_(â„“)H_{G(\mathcal{O}), \mathbb{Z}_{\ell}} \neq H_{G(\mathcal{O})}^{\mathrm{cl}} \otimes \mathbb{Z}_{\ell} in general.)
There is also a pure Galois side description of A_(I,Q_(â„“))\mathfrak{A}_{I, \mathbb{Q}_{\ell}}, known as the unipotent coherent Springer sheaf as defined in [6,82] (see also [32]).
Example 2.1.7. By construction, there is a natural morphism of stacks Loc_(G)rarrB hat(G)\operatorname{Loc}_{G} \rightarrow \mathbb{B} \hat{G} over Z_(â„“)\mathbb{Z}_{\ell}. For a representation of hat(G)\hat{G} on a finite projective Lambda\Lambda-module, regarded as a vector bundle on B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}, let tilde(V)\tilde{V} be its pullback to Loc_(G)ox Lambda\operatorname{Loc}_{G} \otimes \Lambda, and let tilde(V)^(?)in Perf(Loc_(c_(G))ox Lambda)\tilde{V}{ }^{?} \in \operatorname{Perf}\left(\operatorname{Loc}_{c_{G}} \otimes \Lambda\right) be its
where rr and Nt\mathrm{Nt} are maps in the following correspondence:
Hk_(G)=L^(+)G\\LG//L^(+)Glarr^(r)LG//Ad_(sigma)L^(+)Grarr"Nt"LG//Ad_(sigma)LG=B(G).\mathrm{Hk}_{G}=L^{+} G \backslash L G / L^{+} G \stackrel{r}{\leftarrow} L G / \operatorname{Ad}_{\sigma} L^{+} G \xrightarrow{\mathrm{Nt}} L G / \operatorname{Ad}_{\sigma} L G=\mathfrak{B}(G) .
In particular, for two representations VV and WW of hat(G)\hat{G}, there is a morphism
compatible with compositions. Such map was first constructed in [64,77] and (the version for underived Hom spaces) was then extended to Z_(â„“)\mathbb{Z}_{\ell}-coefficient in [70]. It has significant arithmetic applications, as will be explained in Section 3.
Remark 2.1.8. It is likely that Theorem 2.1 .5 can be extended to the tame level by taking the Frobenius-twisted categorical trace of the equivalence from Theorem 1.2.4. On the other hand, as mentioned in Remark 1.2.5, it is important to extend these equivalences to Z_(â„“^(-))\mathbb{Z}_{\ell^{-}} coefficient.
2.2. Global arithmetic Langlands for function fields
Next we turn to global aspects of the arithmetic Langlands correspondence. As mentioned at the beginning, its classical formulation, very roughly speaking, predicts a natural correspondence between the set of (irreducible) Galois representations and the set of (cuspidal) automorphic representations. As in the local case, beyond the GL_(n)\mathrm{GL}_{n} case (which is a theorem by [38]), such a formulation is not easy to be made precise. On the other hand, the global geometric Langlands conjecture from Section 1.3 and philosophy of decategorification/trace suggest that the global arithmetic Langlands can and probably should be formulated as an isomorphism between two vector spaces, arising from the Galois and the automorphic side, respectively. In this subsection, we formulate such a conjecture in the global function field case.
Let F=F_(q)(X)F=\mathbb{F}_{q}(X) be the function field of a geometrically connected smooth projective curve XX over F_(q)\mathbb{F}_{q}. We write eta=Spec F\eta=\operatorname{Spec} F for the generic point of XX and bar(eta)\bar{\eta} for a geometric point over eta\eta. Let |X||X| denote the set of closed points of XX. For v in|X|v \in|X|, let O_(v)\mathcal{O}_{v} denote the complete local ring of XX at vv and F_(v)F_{v} its fractional field. Let O_(F)=prod_(v in|X|)O_(v)\mathbb{O}_{F}=\prod_{v \in|X|} \mathcal{O}_{v} be the integral adèles, and A_(F)=prod_(v in|X|)^(')F_(v)\mathbb{A}_{F}=\prod_{v \in|X|}^{\prime} F_{v} the ring of adèles. For a finite nonempty set of places QQ, let W_(F,Q)W_{F, Q} denote the Weil group of FF, unramified outside QQ.
Let GG be a connected reductive group over FF. Similarly to the local situation, the first step to formulate our global conjecture is the following theorem from [82].
Theorem 2.2.1. Assume that ℓ∤2p\ell \nmid 2 p. The prestack sending aZ_(ℓ)a \mathbb{Z}_{\ell}-algebra AA to the space of (strongly) continuous homomorphisms rho:W_(F,Q)rarr^(c)G(A)\rho: W_{F, Q} \rightarrow{ }^{c} G(A) such that d@rho=(cycl^(-1),pr)d \circ \rho=\left(\mathrm{cycl}^{-1}, \mathrm{pr}\right) is represented by a derived scheme Loc_(c)^(◻)\operatorname{Loc}_{c}^{\square}, , which is a disjoint union of derived affine schemes that are flat and of finite type over Z_(ℓ)\mathbb{Z}_{\ell}. If Q!=O/,Loc_(cG,Q)^(◻)Q \neq \emptyset, \operatorname{Loc}_{c G, Q}^{\square} is quasismooth.
We then define the stack of global Langlands parameters as Loc_(G,Q)=Loc_(_(G),Q)// hat(G)\operatorname{Loc}_{G, Q}=\operatorname{Loc}_{{ }_{G}, Q} / \hat{G}. Similar to the local case (see Example 2.1.7), for a representation of hat(G)_(Lambda)\hat{G}_{\Lambda} on a finite projective Lambda\Lambda-module, regarded as a vector bundle on B hat(G)_(Lambda)\mathbb{B} \hat{G}_{\Lambda}, let tilde(V)\tilde{V} be its pullback to Loc_(G,Q)ox Lambda\operatorname{Loc}_{G, Q} \otimes \Lambda. If VV is the restriction of a representation of (^(c)G)^(S)\left({ }^{c} G\right)^{S} along the diagonal embedding hat(G)rarr(^(c)G)^(S)\hat{G} \rightarrow\left({ }^{c} G\right)^{S}, then there is a natural (strongly) continuous W_(F,Q)^(S)W_{F, Q}^{S}-action on tilde(V)\tilde{V} (see [82, SEcT. 2.4]). For a place vv of FF, let Loc_(_(c)G,v)\operatorname{Loc}_{{ }_{c} G, v} denote the stack of local Langlands parameters for G_(F_(v))G_{F_{v}}. Let
" res ":Loc_(G,Q)rarrprod_(v in Q)Loc_(G,v)\text { res }: \operatorname{Loc}_{G, Q} \rightarrow \prod_{v \in Q} \operatorname{Loc}_{G, v}
denote the map by restricting global parameters to local parameters (induced by the map {:W_(F_(v))rarrW_(F,Q))\left.W_{F_{v}} \rightarrow W_{F, Q}\right). Later on, we will consider the !-pullback of coherent sheaves on prod_(v in Q)Loc_(c,v)\prod_{v \in Q} \operatorname{Loc}_{c, v} along this map.
Remark 2.2.2. (1) In fact, when Q=O/Q=\emptyset, the definition of Loc_(G,Q)\operatorname{Loc}_{G, Q} needs to be slightly modified.
(2) Unlike the local situation, Loc_(G),Q\operatorname{Loc}_{G}, Q has nontrivial derived structure in general (see [82, REMARK 3.4.5]). Let ^(cl)Loc^(G,Q)^(Q){ }^{c l} \operatorname{Loc}^{G, Q}{ }^{Q} denote the underlying classical stack.
(3) A different definition of Loc_(G,Q)oxQ_(â„“)\operatorname{Loc}_{G, Q} \otimes \mathbb{Q}_{\ell} is given by [3].
Next we move to the automorphic side. For simplicity, we assume that GG is split over F_(q)\mathbb{F}_{q} in this subsection. Fix a level, i.e., an open compact subgroup K sub G(O_(F))K \subset G\left(\mathbb{O}_{F}\right). Let QQ be the set of places consisting of those vv such that K_(v)!=G(O_(v))K_{v} \neq G\left(\mathcal{O}_{v}\right). For a finite set SS, let Sht _(K)(G)_((X-Q)^(S))_{K}(G)_{(X-Q)^{S}} denote the ind-Deligne-Mumford stack over (X-Q)^(S)(X-Q)^{S} of the moduli of GG-shtukas on XX with SS-legs in X-QX-Q and KK-level structure. (For example, see [39] for basic constructions and properties of this moduli space.) Its base change along the diagonal map bar(eta)rarr(X-Q)rarr"Delta"(X-Q)^(S)\bar{\eta} \rightarrow(X-Q) \xrightarrow{\Delta}(X-Q)^{S} is denoted by Sht_(K)(G)_(Delta( bar(eta)))\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}. For every representation VV of (^(c)G)^(S)\left({ }^{c} G\right)^{S} on a finite projective Lambda\Lambda-module, the geometric Satake (1.6) (with DD replaced by X-QX-Q and with Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell} allowed) provides a perverse sheaf Sat_(S)(V)\operatorname{Sat}_{S}(V) on Sht_(K)(G)_((X-Q)^(S))\operatorname{Sht}_{K}(G)_{(X-Q)^{S}}. Let C_(c)(Sht_(K)(G)_(Delta( bar(eta))):}C_{c}\left(\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})}\right., Sat {:S_(S)(V))\left.S_{S}(V)\right) denote the (cochain complex of the) total compactly supported cohomology of Sht_(K)(G)_(Delta( bar(eta)))\operatorname{Sht}_{K}(G)_{\Delta(\bar{\eta})} with coefficient in Sat_(S)(V)\operatorname{Sat}_{S}(V). It admits a (strongly) continuous action of W_(F,Q)^(S)W_{F, Q}^{S} (see [34] for the construction of such action at the derived level, based on [67,68][67,68] ), as well as an action of the corresponding global (derived) Hecke algebra (with coefficients in Lambda\Lambda )
Here G(F)\\G(A)//KG(F) \backslash G(\mathbb{A}) / K is regarded as a discrete DM stack over bar(eta)\bar{\eta}, and C_(c)(G(F)\\G(A)//K,Lambda)C_{c}(G(F) \backslash G(\mathbb{A}) / K, \Lambda) denotes its compactly supported cohomology. When Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell}, this is the space of compactly supported functions on G(F)\\G(A)//KG(F) \backslash G(\mathbb{A}) / K.
We will fix a pinning (B,T,e)(B, T, e) of GG and a nondegenerate character psi:F\\Ararr\psi: F \backslash \mathbb{A} \rightarrowZ_(â„“)[mu_(p)]^(xx)\mathbb{Z}_{\ell}\left[\mu_{p}\right]^{\times}, which gives the conjectural equivalence L_(v)\mathbb{L}_{v} as in Conjecture 2.1.3 for every v in Qv \in Q. In particular, corresponding to K_(v)sub G(F_(v))K_{v} \subset G\left(F_{v}\right) there is a conjectural coherent sheaf N_(K_(v))\mathfrak{N}_{K_{v}} (see (2.1)) on Loc_(G,v)\operatorname{Loc}_{G, v}.
Conjecture 2.2.3. There is a natural (W_(F,Q)^(S)xxH_(K,Lambda))\left(W_{F, Q}^{S} \times H_{K, \Lambda}\right)-equivariant isomorphism
We refer to [82, SECT. 4.7] for more general form of the conjecture (where "generalized level structures" are allowed) and examples of such conjecture in various special cases. This conjecture could be regarded a precise form of the global Langlands correspondence for function fields. Namely, it gives a precise recipe to match Galois representations and automorphic representations. (For example, V. Lafforgue's excursion operators are encoded in such isomorphism, see below.) Moreover, such an isomorphism fits in the Arthur-Kottwitz multiplicity formula and at the same time extends such a formula to the integral level and therefore relates to automorphic lifting theories.
The most appealing evidence of this conjecture is the following theorem [40,82], as suggested (at the heuristic level) by Drinfeld as an interpretation of Lafforgue's construction.
with an action of H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}, such that for every finite dimensional Q_(â„“)\mathbb{Q}_{\ell}-representation VV of (^(c)G)^(S)\left({ }^{c} G\right)^{S}, there is a natural (W_(F,Q)^(S)xxH_(K,Q_(â„“)))\left(W_{F, Q}^{S} \times H_{K, \mathbb{Q}_{\ell}}\right)-equivariant isomorphism
We mention that this theorem actually was proved for any GG in [40,82]. In addition, when KK is everywhere hyperspecial, (2.2.4) holds at the derived level by [3].
The isomorphism (2.2.4) induces an action of Gamma(^(cl)Loc^(c)_(G,Q)oxQ_(â„“),O)\Gamma\left({ }^{c l} \operatorname{Loc}^{c}{ }_{G, Q} \otimes \mathbb{Q}_{\ell}, \mathcal{O}\right) on the righthand side. This is exactly the action by V\mathrm{V}. Lafforgue's excursion operators, which induces the decomposition of the right-hand side (in particular, C_(c)(G(F)\\G(A_(F))//K,Q_(â„“))C_{c}\left(G(F) \backslash G\left(\mathbb{A}_{F}\right) / K, \mathbb{Q}_{\ell}\right) ) in terms of semisimple Langlands parameters. As explained [40], over an elliptic Langlands parameter, such an isomorphism is closely related to the Arthur-Kottwitz multiplicity formula. In the case of G=GL_(n)G=\mathrm{GL}_{n}, it gives the following corollary, generalizing [38].
Corollary 2.2.5. Let pi\pi be a cuspidal automorphic representation of GL_(n)\mathrm{GL}_{n}, with the associated irreducible Galois representation rho_(pi):W_(F,Q)rarrGL_(n)(Lambda)\rho_{\pi}: W_{F, Q} \rightarrow \operatorname{GL}_{n}(\Lambda) for some finite extension Lambda//Q_(â„“)\Lambda / \mathbb{Q}_{\ell} and with m_(pi)\mathfrak{m}_{\pi} the corresponding maximal ideal of Gamma(^(cl)Loc^(G),Q ox Lambda,O)\Gamma\left({ }^{c l} \operatorname{Loc}^{G}, Q \otimes \Lambda, \mathcal{O}\right). Then there is an (W_(F,Q)^(S)xxH_(K))\left(W_{F, Q}^{S} \times H_{K}\right)-equivariant isomorphism
In particular, the left-hand side only concentrates in cohomological degree zero.
2.3. Geometric realization of Jacquet-Langlands transfer
The global Langlands correspondence for number fields is far more complicated In fact, there are analytic part of the theory which currently seems not to fit the categorification/decategorification framework. Even if we just restrict to the algebraic/arithmetic part of the theory, there are complications coming from the place at â„“\ell and at oo\infty. In particular, the categorical forms of the local Langlands correspondence at â„“\ell and oo\infty are not yet fully understood.
Nevertheless, in a forthcoming joint work with Emerton and Emerton-Gee [21, 22], we will formulate conjectural Galois theoretical descriptions for the cohomology of Shimura varieties and even cohomology for general locally symmetric space, parallel to Conjecture 2.2.3. In this subsection, we just review a conjecture from [82] on the geometric realization of Jacquet-Langlands transfer via cohomology of Shimura varieties and discuss results from [35,64][35,64] towards this conjecture.
We fix a few notations and assumptions. We fix a prime pp in this subsection.
Let A_(f)=prod_(q)^(')Q_(q)\mathbb{A}_{f}=\prod_{q}^{\prime} \mathbb{Q}_{q} denote the ring of finite adèles of Q\mathbb{Q}, and A_(f)^(p)=prod_(q!=p)^(')Q_(p)\mathbb{A}_{f}^{p}=\prod_{q \neq p}^{\prime} \mathbb{Q}_{p}. We write bar(eta)=Spec bar(Q)\bar{\eta}=\operatorname{Spec} \overline{\mathbb{Q}}, where bar(Q)\overline{\mathbb{Q}} is the algebraic closure of Q\mathbb{Q} in C\mathbb{C}. For a Shimura datum (G,X)(G, X), let mu\mu be the (minuscule) dominant weight of hat(G)\hat{G} (with respect to ( hat(B), hat(T))(\hat{B}, \hat{T}) ) determined by (G,X)(G, X) in the usual way and let V_(mu)V_{\mu} denote the minuscule representation of hat(G)\hat{G} of highest weight mu\mu. Let E sub bar(Q)subCE \subset \overline{\mathbb{Q}} \subset \mathbb{C} be the reflex field of (G,X)(G, X) and write d_(mu)=dim Xd_{\mu}=\operatorname{dim} X. For a level (i.e., an open compact subgroup) K=K_(p)K^(p)sub G(Q_(p))G(A_(f)^(p))K=K_{p} K^{p} \subset G\left(\mathbb{Q}_{p}\right) G\left(\mathbb{A}_{f}^{p}\right), let Sh_(K)(G)\operatorname{Sh}_{K}(G) be the corresponding
Shimura variety of level KK (defined over the reflex field EE ), and let Sh_(K)(G)_( bar(eta))\operatorname{Sh}_{K}(G)_{\bar{\eta}} denote its base change along E rarr bar(Q)E \rightarrow \overline{\mathbb{Q}}. Let vv be a place of EE above pp. By a specialization sp: bar(eta)rarr bar(v)\mathrm{sp}: \bar{\eta} \rightarrow \bar{v}, we mean a morphism from bar(eta)\bar{\eta} to the strict henselianization of O_(E)\mathcal{O}_{E} at vv.
To avoid many complications from Galois cohomology (e.g., the difference between extended pure inner forms and inner forms) and also some complications from geometry (e.g., the relation between Shimura varieties and moduli of Shtukas), we assume that GG is of adjoint type in the rest of this subsection, and refer to [64] for general GG. See also [82] with less restrictions on GG.
Definition 2.3.1. Let GG be a connected reductive group over Q\mathbb{Q}. A prime-to- pp (resp. finitely) trivialized inner form of GG is a GG-torsor beta\beta over Q\mathbb{Q} equipped with a trivialization beta\beta over A_(f)^(p)\mathbb{A}_{f}^{p} (resp. over A_(f)\mathbb{A}_{f} ). Then G^('):=Aut(xi)G^{\prime}:=\operatorname{Aut}(\xi) is an inner form of GG (so the dual group of GG and G^(')G^{\prime} are canonically identified), equipped with an isomorphism theta:G(A_(f)^(p))~=G^(')(A_(f)^(p))\theta: G\left(\mathbb{A}_{f}^{p}\right) \cong G^{\prime}\left(\mathbb{A}_{f}^{p}\right) (resp. {: theta:G(A_(f))~=G^(')(A_(f)))\left.\theta: G\left(\mathbb{A}_{f}\right) \cong G^{\prime}\left(\mathbb{A}_{f}\right)\right).
Now let (G,X)(G, X) and (G^('),X^('))\left(G^{\prime}, X^{\prime}\right) be two Shimura data, with G^(')G^{\prime} a prime-to- pp trivialized inner form of GG. Via theta\theta, one can transport K^(p)sub G(A_(f)^(p))K^{p} \subset G\left(\mathbb{A}_{f}^{p}\right) to an open compact subgroup K^('p)subG^(')(A_(f)^(p))K^{\prime p} \subset G^{\prime}\left(\mathbb{A}_{f}^{p}\right). We identify the prime-to- pp (derived) Hecke algebra H_(K^(p),Lambda)H_{K^{p}, \Lambda} (defined in the same way as in (2.5)) with H_(K^('p),Lambda)H_{K^{\prime p}, \Lambda} and simply write them as H_(K^(p),Lambda)H_{K^{p}, \Lambda}. Let K_(p)^(')subG^(')(Q_(p))K_{p}^{\prime} \subset G^{\prime}\left(\mathbb{Q}_{p}\right) be an open compact subgroup and write K^(')=K_(p)^(')K^('p)K^{\prime}=K_{p}^{\prime} K^{\prime p} for the corresponding level.
We fix a quasisplit inner form G_(Q_(p))^(**)G_{\mathbb{Q}_{p}}^{*} of G_(Q_(p))G_{\mathbb{Q}_{p}} and G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime} equipped with a pinning (B_(Q_(p))^(**),T_(Q_(p))^(**),e^(**))\left(B_{\mathbb{Q}_{p}}^{*}, T_{\mathbb{Q}_{p}}^{*}, e^{*}\right), and realize G_(Q_(p))G_{\mathbb{Q}_{p}} as J_(b)J_{b} and G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime} as J_(b^('))J_{b^{\prime}} for b,b in B(G_(Q_(p))^(**))b, b \in B\left(G_{\mathbb{Q}_{p}}^{*}\right). Under our assumption that GG and G^(')G^{\prime} are adjoint, such b,b^(')b, b^{\prime} exist and are unique. Then we have the conjectural coherent sheaf U_(K_(p),Lambda)\mathfrak{U}_{K_{p}, \Lambda} and H_(K_(p)^('),Lambda)\mathscr{H}_{K_{p}^{\prime}, \Lambda} as in (2.1) on the stack Loc _(G,p)ox Lambda{ }_{G, p} \otimes \Lambda of local Langlands parameters for G_(Q_(p))^(**)G_{\mathbb{Q}_{p}}^{*} over Lambda\Lambda.
Conjecture 2.3.2. For every choice of specialization map sp: bar(eta)rarr bar(v)\mathrm{sp}: \bar{\eta} \rightarrow \bar{v}, there is a natural map
compatible with compositions. In particular, there is an ( E_(1^(-))E_{1^{-}})algebra homomorphism
compatible with (2.6). In addition, the induced action
{:(2.8)H_(K_(p),Lambda)~=^((2.2))R End(U_(K_(p),Lambda))rarr R End(( widetilde(V_(mu)))oxU_(K_(p),Lambda))rarr"S"REnd_(H_(K^(p),Lambda))(C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)):}\begin{equation*}
H_{K_{p}, \Lambda} \stackrel{(2.2)}{\cong} R \operatorname{End}\left(\mathfrak{U}_{K_{p}, \Lambda}\right) \rightarrow R \operatorname{End}\left(\widetilde{V_{\mu}} \otimes \mathfrak{U}_{K_{p}, \Lambda}\right) \xrightarrow{S} R \operatorname{End}_{H_{K^{p}, \Lambda}}\left(C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right)\right) \tag{2.8}
\end{equation*}
coincides with the natural Hecke action of H_(K_(p),Lambda)H_{K_{p}, \Lambda} on C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right) (and therefore is independent of the specialization map sp\mathrm{sp} ).
This conjecture would be a consequence of a Galois theoretic description of C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right) similar to Conjecture 2.2.3, but its formulation does not require the existence of the stack of global Langlands parameters for Q\mathbb{Q}. In any case, a step towards a Galois-theoretical description of C_(c)(Sh_(K)(G)_( bar(eta)),Lambda)C_{c}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right) might require Conjecture 2.3.2 as an
input. We also remark that as in the function field case, there is a more general version of such conjecture in [82, SECT. 4.7], allowing "generalized level structures," so that the cohomology of Igusa varieties could appear.
The following theorem verifies the conjecture in special cases.
Theorem 2.3.3. Suppose that the Shimura data (G,X)(G, X) and (G^('),X^('))\left(G^{\prime}, X^{\prime}\right) are of abelian type, with G^(')G^{\prime} a finitely trivialized inner form of GG. Suppose that G_(Q_(p))G_{\mathbb{Q}_{p}} is unramified (and therefore so is G_(Q_(p))^(')G_{\mathbb{Q}_{p}}^{\prime} ).
(1) The map (2.6) (and therefore (2.7)) exists when Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell} and K_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right) and K_(p)^(')subG^(')(Q_(p))K_{p}^{\prime} \subset G^{\prime}\left(\mathbb{Q}_{p}\right) are parahoric subgroups (in the sense of Bruhat-Tits).
(2) If K_(p)K_{p} is hyperspecial, then the map (2.6) (and therefore (2.7)) exists when Lambda=Z_(â„“)\Lambda=\mathbb{Z}_{\ell}, at least for underived Hom spaces. In addition, the action of H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}} on H_(c)^(**)(Sh_(K)(G)_( bar(eta)),Lambda)H_{c}^{*}\left(\operatorname{Sh}_{K}(G)_{\bar{\eta}}, \Lambda\right) via (2.8) coincides with the natural action of H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}.
Part (1) is proved in [35,64]. The proof contains two ingredients. One is the construction of physical correspondences between mod pp fibers of Sh_(K)(G)\mathrm{Sh}_{K}(G) and Sh_(K^('))(G^('))\mathrm{Sh}_{K^{\prime}}\left(G^{\prime}\right) by [64] (this is where we currently need to assume that GG and G^(')G^{\prime} are unramified at pp ). The other ingredient is Theorem 2.1.5 (and therefore requires Lambda=Q_(â„“)\Lambda=\mathbb{Q}_{\ell} ). When K_(p)K_{p} is hypersepcial, one can work with Z_(â„“)\mathbb{Z}_{\ell}-coefficient, as (the underived version of) (2.4) exists for Z_(â„“)\mathbb{Z}_{\ell}-coefficient thanks to [70]. In fact, in this case one can allow nontrivial local systems on the Shimura varieties (see [70]). The last statement is known as the S=TS=T for Shimura varieties. The case when d_(mu)=dim Sh_(K)(G)=0d_{\mu}=\operatorname{dim} \operatorname{Sh}_{K}(G)=0 is contained in [64]. The general case is proved in [63,74] using foundational works from [25,59][25,59].
3. APPLICATIONS TO ARITHMETIC GEOMETRY
Besides the previously mentioned directly applications of (ideas from) geometric Langlands to the classical Langlands program, we discuss some further arithmetic applications, mostly related to Shimura varieties and based on the author's works. We shall mention that there are many other remarkable applications of (ideas of) geometric Langlands to arithmetic problems, such as [28,31,44,66,71][28,31,44,66,71], to name a few.
We use notations from Section 2.3 for Shimura varieties (but we do not assume that GG is of adjoint type in this subsection). Let (G,X)(G, X) be a Shimura datum and KK a chosen level with K_(p)=E(Z_(p))K_{p}=\mathscr{E}\left(\mathbb{Z}_{p}\right) for some parahoric group scheme E\mathscr{E} (in the sense of Bruhat-Tits) of G_(Q_(p))G_{\mathbb{Q}_{p}} over Z_(p)\mathbb{Z}_{p}. Then for a place vv of EE over pp, a local model diagram is a correspondence of quasiprojective schemes over O_(E_(v))\mathcal{O}_{E_{v}},
The original construction of local models is based on realization of a parahoric group scheme as (the neutral connected component of) the stabilizer group of a self-dual lattice chain in a vector space (over a division algebra over FF ) with a bilinear form, e.g., see [57] for a survey and references. This approach is somehow ad hoc and is limited the so-called (P)EL (local) Shimura data. A new approach, based on the construction of an Z_(p)\mathbb{Z}_{p}-analogue of the stack Hk_(g,D)\mathrm{Hk}_{\boldsymbol{g}, D} from Section 1.2, was systematically introduced in [58] (under the tameness assumption of GG which was later lifted in [46,50]). In [58] the construction of such a Z_(p)\mathbb{Z}_{p}-analogue (or rather the corresponding Beilinson-Drinfeld-type affine Grassmannian Gr Z_(p)\operatorname{Gr} \mathscr{Z}_{\mathbb{p}} over Z_(p)\mathbb{Z}_{p} ) is based on the construction of certain "two dimensional parahoric" group scheme tilde(E)\tilde{\mathscr{E}} over Z_(p)[Ï–]\mathbb{Z}_{p}[\varpi] whose restriction along Z_(p)[Ï–]rarr"Ï–|->p"Z_(p)\mathbb{Z}_{p}[\varpi] \xrightarrow{\varpi \mapsto p} \mathbb{Z}_{p} recovers E\mathscr{E}. (See [81] for a survey.) A more direct construction of a different pp-adic version of such affine Grassman-
model is defined as the flat closure of the Schubert variety in the generic fiber corresponding to mu\mu. In addition, the recent work [1] shows that the two constructions agree. The following theorem from [1] is the most up-to-date result on the existence of local models and about their properties.
Theorem 3.1.1. Let GG be a connected reductive group over a p-adic field FF. Except the odd unitary case when p=2p=2 and triality case when p=3p=3, for every parahoric group scheme E\mathcal{E} of GG over O\mathcal{O}, and a conjugacy class of minuscule cocharacters mu\mu of GG defined over a finite extension E//FE / F of FF, there is a normal flat projective scheme M_(G,mu)^(loc)M_{\mathscr{G}, \mu}^{\mathrm{loc}} over O_(E)\mathcal{O}_{E}, equipped with a E_(O_(E))\mathscr{E}_{\mathcal{O}_{E}}-action such that M_(Q,mu)^(loc)ox EM_{\mathscr{Q}, \mu}^{\mathrm{loc}} \otimes E is G_(E)G_{E}-equivariantly isomorphic to the partial flag variety Fâ„“_(mu)\mathscr{F} \ell_{\mu} of G_(E)G_{E} corresponding to mu\mu, and that M_(G)^(loc)oxk_(E)M_{\mathscr{G}}^{\mathrm{loc}} \otimes k_{E} is (Goxk_(E))\left(\mathscr{G} \otimes k_{E}\right)-equivariantly isomorphic to the (canonical deperfection of the) union over the mu\mu-admissible set of Schubert varieties in LG//L^(+)Eoxk_(E)L G / L^{+} \mathcal{E} \otimes k_{E}. In addition, M_(G)^("loc ")M_{\mathscr{G}}^{\text {loc }} is normal, Cohen-Macaulay and each of its geometric irreducible components in its special fiber is normal and Cohen-Macaulay.
We end this subsection with a few remarks.
Remark 3.1.2. (1) Once the local model diagram (3.1) is established, this theorem also gives the corresponding properties of the integral models of Shimura varieties.
(2) A key ingredient in the study of special fibers of local models is the coherence conjecture by Pappas-Rapoport [56], proved in [75] (and the proof uses the idea of fusion).
(3) One important motivation/application of the theory of local models is the Haines-Kottwitz conjecture [29], which predicts certain central element in the parahoric Hecke algebra H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}} should be used as the test function in the trace formula computing the Hasse-Weil zeta function of Sh_(K)(G)\mathrm{Sh}_{K}(G). As mentioned in Section 1.2, this conjecture motivated Gaitsgory's central sheaf construction (1.9). With the local Hecke stack HkZ_(,)Z_(p)\mathrm{Hk} \mathscr{Z}_{,} \mathbb{Z}_{p} over Z_(p)\mathbb{Z}_{p} constructed (either the version from [58] or from [59]), one can mimic the construction (1.9) in mixed characteristic to solve the Kottwitz conjecture. Again, see [1] for the up-to-date result.
3.2. The congruence relation
We use notations and (for simplicity) keep assumptions from Section 2.3 regarding Shimura varieties. Let (G,X)(G, X) be a Shimura datum abelian type, and let KK be a level such that K_(p)K_{p} is hyperspecial. Let v∣pv \mid p be the place of EE. Then Sh_(K)(G)\operatorname{Sh}_{K}(G) has a canonical integral model S_(K)\mathscr{S}_{K} defined over O_(E,(v))\mathcal{O}_{E,(v)} [37]. Let bar(S)_(K)\overline{\mathscr{S}}_{K} be its mod pp fiber, which is a smooth variety defined over the residue field k_(v)k_{v} of vv. Let sigma_(v)\sigma_{v} denote the geometric Frobenius in Gamma_(k_(v))\Gamma_{k_{v}}. Theorem 2.3.3
consequences.
The congruence relation conjecture (also known as the Blasius-Rogawski conjecture), generalizing the classical Eichler-Shimura congruence relation Frob _(p)=T_(p)+V_(p){ }_{p}=T_{p}+V_{p} for modular curves, predicts that in the Chow group of bar(S)_(K)xx bar(S)_(K)\overline{\mathscr{S}}_{K} \times \overline{\mathscr{S}}_{K}, the Frobenius endomorphism of bar(S)_(K)\overline{\mathscr{S}}_{K} satisfies a polynomial whose coefficients are mod pp reduction of certain Hecke correspondences. Theorem 2.3.3, together with [65, SECT. 6.3], implies this conjecture at the level of cohomology.
For every representation VV of ^(c)(G_(Q_(p))){ }^{c}\left(G_{\mathbb{Q}_{p}}\right), its character chi_(V)\chi_{V} (regarded as a hat(G)\hat{G}-invariant function on {:^(c)G|_(d=(p,sigma_(p))))\left.\left.{ }^{c} G\right|_{d=\left(p, \sigma_{p}\right)}\right) gives an element h_(V)inH_(G(Z_(p)))^(cl)h_{V} \in H_{G\left(\mathbb{Z}_{p}\right)}^{\mathrm{cl}} via the Satake isomorphism (1.3).
Indeed, by [65, sEct. 6.3], such an equality holds with h_(chi_(^^^(i)V))h_{\chi_{\wedge^{i} V}} replaced by S(chi_(^^^(i)V))S\left(\chi_{\wedge^{i} V}\right), where SS is from Theorem 2.3.3 (1). Then part (2) of that theorem allows one to replace S(chi_(^^^(i)V))S\left(\chi_{\wedge^{i} V}\right) by h_(chi_(^^^(i)V))h_{\chi_{\wedge^{i} V}}. This approach to (3.2) is the Shimura variety analogue of V. Lafforgue's approach to the Eichler-Shimura relation for Sht_(K)(G)\operatorname{Sht}_{K}(G) [39]. Traditionally, there is another approach to the congruence relation conjecture for Shimura varieties by directly studying reduction mod pp of Hecke operators, starting from [24] for the Siegel modular variety case.
See [45] for the latest progress and related references. This approach would give (3.2) at the level of algebraic correspondences.
Now suppose (G,X)=(Res_(F^(+)//Q)(G_(0))_(F^(+)),prod_(varphi:F^(+)rarrR)X_(0))(G, X)=\left(\operatorname{Res}_{F^{+} / \mathbb{Q}}\left(G_{0}\right)_{F^{+}}, \prod_{\varphi: F^{+} \rightarrow \mathbb{R}} X_{0}\right), where (G_(0),X_(0))\left(G_{0}, X_{0}\right) is a Shimura datum and F^(+)F^{+}is a totally real field. As before, let pp be a prime such that K_(p)K_{p} is hyperspecial. In particular, pp is unramified in F^(+)F^{+}. In addition, for simplicity we assume that G_(0,Q_(p))G_{0, \mathbb{Q}_{p}} is split (so for a place vv of EE above p,E_(v)=Q_(p)p, E_{v}=\mathbb{Q}_{p} ). We let F\mathbb{F} denote an algebraic closure of F_(p)\mathbb{F}_{p}. Let {w_(i)}_(i)\left\{w_{i}\right\}_{i} be the set of primes of F^(+)F^{+}above pp, and let k_(i)k_{i} denote the residue field of w_(i)w_{i}. For each ii, we also fix an embedding rho_(i):k_(i)rarrF\rho_{i}: k_{i} \rightarrow \mathbb{F}. Then there is a natural map
where S_(f_(i))\mathbb{S}_{f_{i}} is the permutation group on f_(i)f_{i} letters. Together with Theorem 2.3.3, one obtains the following result [64].
Theorem 3.2.2. There is an action of prod_(i)(Z^(f_(i))><|S_(f_(i)))\prod_{i}\left(\mathbb{Z}^{f_{i}} \rtimes \mathbb{S}_{f_{i}}\right) on H_(c)^(**)( bar(S)_(K, bar(F)),Z_(â„“))H_{c}^{*}\left(\overline{\mathscr{S}}_{K, \overline{\mathbb{F}}}, \mathbb{Z}_{\ell}\right) such that action of sigma_(p)\sigma_{p} factors as sigma_(p)=prod_(i)sigma_(p,i)\sigma_{p}=\prod_{i} \sigma_{p, i}, where sigma_(p,i)=((1,0,dots,0),(12 cdotsf_(i)))inZ^(f_(i))><|S_(f_(i))\sigma_{p, i}=\left((1,0, \ldots, 0),\left(12 \cdots f_{i}\right)\right) \in \mathbb{Z}^{f_{i}} \rtimes \mathbb{S}_{f_{i}}. Each sigma_(p,i)^(f_(i))\sigma_{p, i}^{f_{i}} satisfies a polynomial equation similar to (3.2).
This theorem gives some shadow of the plectic cohomology conjecture of NekovářScholl [54].
3.3. Generic Tate cycles on mod p\boldsymbol{p} fibers of Shimura varieties
In [64], we applied Theorem 2.3.3 to verify "generic" cases of Tate conjecture for the mod p\bmod p fibers of many Shimura varieties. We use notations and (for simplicity) keep assumptions from Section 3.2. Let ( bar(S)_(K, bar(k_(v))))^("pf ")\left(\overline{\mathscr{S}}_{K, \overline{k_{v}}}\right)^{\text {pf }} denote the perfection of bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}} (i.e., regard it as a perfect presheaf over {:Aff((kf)/( bar(k)_(v))))\left.\mathbf{A f f} \frac{\mathrm{kf}}{\bar{k}_{v}}\right), then by attaching to every point of bar(S)_(K, bar(k))\overline{\mathscr{S}}_{K, \bar{k}} an FF-isocrystal with GG-structure (see [37,64][37,64] ), one can define the so-called Newton map
Then the Newton stratification of B(G_(Q_(p)))_( bar(k_(v)))\mathfrak{B}\left(G_{\mathbb{Q}_{p}}\right)_{\overline{k_{v}}} (see Section 2.1) induces a stratification of bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}} by locally closed subvarieties. It is known that the image of Nt\mathrm{Nt} contains a unique basic element bb and the corresponding subvarieties in bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}} is closed, called the basic Newton stratum, and denoted by bar(S)_(b)\overline{\mathscr{S}}_{b}.
Let mm be the order of the action of the geometric Frobenius sigma_(p)\sigma_{p} on X∙( hat(T))\mathbb{X} \bullet(\hat{T}). Let
For a representation VV of hat(G)_(Q_(ℓ))\hat{G}_{\mathbb{Q}_{\ell}} and lambda inX∙( hat(T))\lambda \in \mathbb{X} \bullet(\hat{T}), let V(lambda)V(\lambda) denote the lambda\lambda-weight subspace of VV (with respect to hat(T)\hat{T} ), and let
We are in particular interested in the condition V_(mu)^("Tate ")!=0V_{\mu}^{\text {Tate }} \neq 0. As explained in the introduction of [64], under the conjectural Galois theoretic description of the cohomology of the
Shimura varieties (analogous to Conjecture 2.2.3), for a Hecke module pi_(f)\pi_{f} whose Satake parameter at pp is general enough, certain multiple a(pi_(f))a\left(\pi_{f}\right) of the dimension of this vector space should be equal to the dimension of the space of Tate classes in the pi_(f)\pi_{f}-component of the middle dimensional compactly-supported cohomology of bar(S)_(K, bar(k_(v)))\overline{\mathscr{S}}_{K, \overline{k_{v}}}. In addition, this space is usually large. For example, in the case GG is an odd (projective) unitary group of signature (i,n-i)(i, n-i) over a quadratic imaginary field, the dimension of this space at an inert prime is (((n+1)/(2))/(i))\binom{\frac{n+1}{2}}{i}.
For a (not necessarily irreducible) algebraic variety ZZ of dimension dd over an algebraically closed field, let H_(2d)^(BM)(Z)(-d)H_{2 d}^{\mathrm{BM}}(Z)(-d) denote the (-d)(-d)-Tate twist of the top degree BorelMoore homology, which is the vector space spanned by the irreducible components of ZZ. Now let XX be a smooth variety of dimension d+rd+r defined over a finite field kk of qq elements, and let Z subeX_( bar(k))Z \subseteq X_{\bar{k}} be a (not necessarily irreducible) projective subvariety of dimension dd. There is the cycle class map
Theorem 3.3.1. We write d_(mu)=dim X=2dd_{\mu}=\operatorname{dim} X=2 d and r=dim V_(mu)^("Tate ")r=\operatorname{dim} V_{\mu}^{\text {Tate }}.
(1) The basic Newton stratum bar(S)_(b)\overline{\mathscr{S}}_{b} of bar(S)_(K, bar(k)_(v))\overline{\mathscr{S}}_{K, \bar{k}_{v}} is pure of dimension d. In particular, dd is always an integer. In addition, there is an H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}-equivariant isomorphism
restricted to H_(2d)^(BM)( bar(S)_(b))[pi_(f)]H_{2 d}^{\mathrm{BM}}\left(\overline{\mathscr{S}}_{b}\right)\left[\pi_{f}\right] is injective if the Satake parameter of pi_(f,p)\pi_{f, p} (the component of pi_(f)\pi_{f} at pp ) is V_(mu)V_{\mu}-general.
(3) Assume that Sh_(K)(G)\mathrm{Sh}_{K}(G) is (essentially) a quaternionic Shimura variety or a Kottwitz arithmetic variety. Then the pi_(f)\pi_{f}-isotypical component of the cycle class map is surjective to T_(â„“)^(d) bar((S_(K)))[pi_(f)]T_{\ell}^{d} \overline{\left(\mathscr{S}_{K}\right)}\left[\pi_{f}\right] if the Satake parameter of pi_(f,p)\pi_{f, p} is strongly V_(mu)V_{\mu}-general. In particular, the Tate conjecture holds for these pi_(f)\pi_{f}.
Remark 3.3.2. (1) For a representation VV of hat(G)\hat{G}, the definitions of " VV-general" and "strongly VV-general" Satake parameters can be found in [64, DEFINITION 1.4.2]. Regular semisimple elements in ^(c)G|_(d=(p,sigma_(p)))\left.{ }^{c} G\right|_{d=\left(p, \sigma_{p}\right)} are always VV-general, but not the converse. See [64, remark 1.4.3].
(2) Some special cases of the theorem were originally proved in [33,60][33,60].
The proof of this theorem relies on several different ingredients. Via the RapoportZink uniformization of the basic locus of a Shimura variety, part (2) can be reduced a question about irreducible components of certain affine Deligne-Lusztig varieties, which was studied in [64, §3]. The most difficult is part (2), which we proved by calculating the intersection numbers among all dd-dimensional cycles in bar(S)_(b)\overline{\mathscr{S}}_{b}. These numbers can be encoded in an r xx rr \times r matrix with entries in H_(K_(p))^(cl)H_{K_{p}}^{\mathrm{cl}}. In general, it seems hopeless to calculate this matrix directly and explicitly. However, this matrix can be understood as the composition of certain morphisms in Coh(Loc_(cG,p)^(ur))\operatorname{Coh}\left(\operatorname{Loc}_{c G, p}^{\mathrm{ur}}\right). Namely, first we realize G^(')(Q)\\G^(')(A)//KG^{\prime}(\mathbb{Q}) \backslash G^{\prime}(\mathbb{A}) / K as a Shimura set with mu^(')=0\mu^{\prime}=0 its Shimura cocharacter. Then using Theorem 2.3.3 (and the Satake isomorphism (2.3)), this matrix can be calculated as
Then one needs to determine when this pairing is nondegenerate, which itself is an interesting question in representation theory, whose solution relies on the study of the Chevellay's restriction map for vector-valued functions. The determinant of this matrix was calculated in [65]. Finally, part (3) was proved by comparing two trace formulas, the Lefschetz trace formula for GG and the Arthur-Selberg trace formula for G^(')G^{\prime}.
Example 3.3.3. Let G=U(1,2r)G=\mathrm{U}(1,2 r) be the unitary group ^(7){ }^{7} of (2r+1)(2 r+1)-variables associated to an imaginary quadratic extension E//QE / \mathbb{Q}, whose signature is (1,2r)(1,2 r) at infinity. It is equipped with a standard Shimura datum, giving a Shimura variety (after fixing a level K sub G(A_(f))K \subset G\left(\mathbb{A}_{f}\right) ). In particular, if r=1r=1, this is (essentially) the Picard modular surface. Let pp be a prime inert in EE such that K_(p)K_{p} is hyperspecial. In this case bar(S)_(b)\overline{\mathscr{S}}_{b} is a union of certain Deligne-Lusztig varieties, parametrized by G^(')(Q)\\G^(')(A_(f))//KG^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / K, where G^(')=U(0,2r+1)G^{\prime}=\mathrm{U}(0,2 r+1) that is isomorphic to GG at all finite places. The intersection patterns of these cycles inside bar(S)_(b)\overline{\mathscr{S}}_{b} were (essentially) given in [61] but the intersection numbers between these cycles are much harder to compute. In fact, we do not know how to compute them directly for general rr, except applying Theorem 2.3.3 to this case. (The case r=1r=1 can be handled directly.)
We have hat(G)=GL_(2r+1)\hat{G}=\mathrm{GL}_{2 r+1} on which sigma_(p)\sigma_{p} acts as A|->J(A^(T))^(-1)JA \mapsto J\left(A^{T}\right)^{-1} J, where JJ is the antidiagonal matrix with all entries along the antidiagonal being 1 . The representation V_(mu)V_{\mu} is the standard representation of GL_(2r+1)\mathrm{GL}_{2 r+1}. One checks that dim V_(mu)^("Tate ")=1\operatorname{dim} V_{\mu}^{\text {Tate }}=1 (which is consistent with the above mentioned parameterization of irreducible components of bar(S)_(b)\overline{\mathscr{S}}_{b} by {:G^(')(Q)\\G^(')(A_(f))//K)\left.G^{\prime}(\mathbb{Q}) \backslash G^{\prime}\left(\mathbb{A}_{f}\right) / K\right). We identify the weight lattice of hat(G)\hat{G} as Z^(2r+1)\mathbb{Z}^{2 r+1} as usual. Then Hom_(Coh(Loc_(c)^(ur):})^(ur)(O,( widetilde(V_(mu))))\operatorname{Hom}_{\operatorname{Coh}\left(\operatorname{Loc}_{c}^{\mathrm{ur}}\right.}^{\mathrm{ur}}\left(\mathcal{O}, \widetilde{V_{\mu}}\right) is a free rank one module over Hom_(Coh(Loc_(C,p)^(ur):})^(ur)(O,O)=H_(K_(p))^(cl)oxQâ„“\operatorname{Hom}_{\operatorname{Coh}\left(\operatorname{Loc}_{C, p}^{\mathrm{ur}}\right.}^{\mathrm{ur}}(\mathcal{O}, \mathcal{O})=H_{K_{p}}^{\mathrm{cl}} \otimes \mathbb{Q} \ell. Then a generator a_("in ")\mathbf{a}_{\text {in }} induces an H_(K,Q_(â„“))H_{K, \mathbb{Q}_{\ell}}-equivariant homomorphism
This is not an adjoint group so the example is not consistent with our assumption. But it is more convenient for the discussion here. The computations are essentially the same.
realizing the cycle class map of bar(S)_(b)\overline{\mathscr{S}}_{b} (up to a multiple). The module Hom_(Coh(Loc)^(c_(G,p))^(ur)(( widetilde(V_(mu))),O)\operatorname{Hom}_{\operatorname{Coh}(\operatorname{Loc}}^{c_{G, p}}{ }^{\mathrm{ur}}\left(\widetilde{V_{\mu}}, \mathcal{O}\right) is also free of rank one over H_(K_(p),Q_(â„“))H_{K_{p}, \mathbb{Q}_{\ell}}. For a chosen generator a_("out ")\mathbf{a}_{\text {out }}, the composition
calculates the intersection matrix of those cycles from the irreducible components of bar(S)_(b)\overline{\mathscr{S}}_{b}.
The element h:=a_("out ")@a_("in ")inH_(K_(p),Q_(â„“))h:=\mathbf{a}_{\text {out }} \circ \mathbf{a}_{\text {in }} \in H_{K_{p}, \mathbb{Q}_{\ell}} was explicitly computed in [65, EXAMPLE 6.4.2] (up to obvious modification and also via the Satake isomorphism (1.4)). Namely,
Here, T_(p,j)=1_(K_(p)lambda_(j)(p)K_(p))T_{p, j}=1_{K_{p} \lambda_{j}(p) K_{p}}, with lambda_(i)=(1^(i),0^(2r-2i+1),(-1)^(i))\lambda_{i}=\left(1^{i}, 0^{2 r-2 i+1},(-1)^{i}\right), and [[n],[m]]_(t)\left[\begin{array}{l}n \\ m\end{array}\right]_{t} is the tt-analogue of the binomial coefficient given by
[0]_(t)=1,quad[n]_(t)=(t^(n)-1)/(t-1),quad[n]_(t)!=[n]_(t)[n-1]_(t)cdots[1]_(t),quad[[n],[m]]_(t)=([n]_(t)!)/([n-m]_(t)![m]_(t)!)[0]_{t}=1, \quad[n]_{t}=\frac{t^{n}-1}{t-1}, \quad[n]_{t}!=[n]_{t}[n-1]_{t} \cdots[1]_{t}, \quad\left[\begin{array}{c}
n \\
m
\end{array}\right]_{t}=\frac{[n]_{t}!}{[n-m]_{t}![m]_{t}!}
In other words, the intersection matrix of cycles in bar(S)_(b)\overline{\mathscr{S}}_{b} in this case is calculated by the Hecke operator (3.3).
On interesting consequence is this computation is the following consequence on the intersection theory of the finite Deligne-Lusztig varieties, for which we do not know a direct proof. Let WW be a (2r+1)(2 r+1)-dimensional nondegenerate hermitian space over F_(p^(2))\mathbb{F}_{p^{2}}. Consider the following rr-dimensional Deligne-Lusztig variety
DL_(r):={H sub W" of dimension "r∣H sube(H^((p)))^(_|_)}\mathrm{DL}_{r}:=\left\{H \subset W \text { of dimension } r \mid H \subseteq\left(H^{(p)}\right)^{\perp}\right\}
where H^((p))H^{(p)} the pullback of HH along the Frobenius. Let H\mathscr{H} denote the corresponding universal subbundle of rank rr. Let E=H^((p))ox((H^((p)))^(_|_)//H)\mathcal{E}=\mathscr{H}^{(p)} \otimes\left(\left(\mathscr{H}^{(p)}\right)^{\perp} / \mathscr{H}\right). Then we have
3.4. The Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives
Let MM be a rational pure Chow motive of weight -1 over a number field FF. The Beilinson-Bloch-Kato conjecture, which is a far reaching generalization of the Birch and Swinnerton-Dyer conjecture, predicts deep relations between certain algebraic, analytic, and cohomological invariants attached to MM :
the rational Chow group CH(M)^(0)\mathrm{CH}(M)^{0} of homologically trivial cycles of MM;
the LL-function L(s,M)L(s, M) of MM;
the Bloch-Kato Selmer group H_(f)^(1)(F,H_(â„“)(M))H_{f}^{1}\left(F, H_{\ell}(M)\right) of the â„“\ell-adic realization H_(â„“)(M)H_{\ell}(M) of MM.
The Beilinson-Bloch conjecture predicts an equality
dim_(Q)CH(M)^(0)=ord_(s=0)L(s,M)\operatorname{dim}_{\mathbb{Q}} \mathrm{CH}(M)^{0}=\operatorname{ord}_{s=0} L(s, M)
between the dimension of CH(M)^(0)\mathrm{CH}(M)^{0} and the vanishing order of the LL-function at the central point, while the Bloch-Kato conjecture predicts
This conjecture seems to be completely out of reach at the moment. For example, for a general motive it is still widely open whether the LL-function has a meromorphic continuation to the whole complex plane so that the vanishing order of L(s,M)L(s, M) at s=0s=0 makes sense. (This would follow from the Galois-to-automorphic direction of the Langlands correspondence for number fields.) Despite this, there have been many works testing this conjecture in various special cases, mostly for motives MM of small rank. In the work [49], we verify certain cases of the above conjecture for Rankin-Selberg motives, which consist of a sequence of motives of arbitrarily large rank.
We assume that F//F^(+)F / F^{+}is a (nontrivial) CM extension with F^(+)F^{+}totally real in the sequel.
Theorem 3.4.1. Let A_(1),A_(2)A_{1}, A_{2} be two elliptic curves over F^(+)F^{+}. Assume that
(1) End_( bar(F))A_(i)=Z\operatorname{End}_{\bar{F}} A_{i}=\mathbb{Z} and Hom_( bar(F))(A_(1),A_(2))=0\operatorname{Hom}_{\bar{F}}\left(A_{1}, A_{2}\right)=0;
(2) Sym^(n-1)A_(1)\operatorname{Sym}^{n-1} A_{1} and Sym^(n)A_(2)\operatorname{Sym}^{n} A_{2} are modular;
(3) F^(+)!=QF^{+} \neq \mathbb{Q} if n >= 3n \geq 3.
Under these assumption, if L(n,Sym^(n-1)A_(1)xxSym^(n)A_(2))!=0L\left(n, \operatorname{Sym}^{n-1} A_{1} \times \operatorname{Sym}^{n} A_{2}\right) \neq 0, then for almost â„“\ell,
Here V_(â„“)(A_(i))V_{\ell}\left(A_{i}\right) denotes the rational Tate module of A_(i)A_{i} as usual.
This theorem is in fact a consequence of a more general result concerning BlochKato Selmer groups of Galois representations associated to certain Rankin-Selberg automorphic representations, which we now discuss.
Recall that for an irreducible regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation Pi\Pi of GL_(n)(A_(F))\mathrm{GL}_{n}\left(\mathbb{A}_{F}\right), one can attach a compatible system of Galois representations rho_(Pi,lambda):Gamma_(F)rarrGL_(n)(E_(lambda))\rho_{\Pi, \lambda}: \Gamma_{F} \rightarrow \operatorname{GL}_{n}\left(E_{\lambda}\right), where E subCE \subset \mathbb{C} is a large enough number field and lambda\lambda is a prime of EE (see [16]). Such EE is called a strong coefficient field of Pi\Pi, which in the situation considered below can be taken as the number field generated by Hecke eigenvalues of Pi\Pi.
Theorem 3.4.2. Suppose that F^(+)!=QF^{+} \neq \mathbb{Q} if n >= 3n \geq 3. Let Pi_(n)\Pi_{n} (resp. {:Pi_(n+1))\left.\Pi_{n+1}\right) be an RACSDC automorphic representation of GL_(n)(A_(F))\mathrm{GL}_{n}\left(\mathbb{A}_{F}\right) (resp. GL_(n+1)(A_(F))\mathrm{GL}_{n+1}\left(\mathbb{A}_{F}\right) ) with trivial infinitesimal character. Let E subeCE \subseteq \mathbb{C} be a strong coefficient field for both Pi_(n)\Pi_{n} and Pi_(n+1)\Pi_{n+1}. Let lambda\lambda be an admissible prime of EE with respect to Pi:=Pi_(0)xxPi_(1)\Pi:=\Pi_{0} \times \Pi_{1}. Let rho_(Pi,lambda):=rho_(Pi_(n),lambda)ox_(E_(lambda))rho_(Pi_(n+1),lambda)\rho_{\Pi, \lambda}:=\rho_{\Pi_{n}, \lambda} \otimes_{E_{\lambda}} \rho_{\Pi_{n+1}, \lambda}.
(1) If the Rankin-Selberg LL-value L((1)/(2),Pi)!=0^(8)L\left(\frac{1}{2}, \Pi\right) \neq 0^{8}, then H_(f)^(1)(F,rho_(Pi,lambda)(n))=0H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right)=0.
(2) If certain element Delta_(lambda)inH_(f)^(1)(F,rho_(Pi,lambda)(n))\Delta_{\lambda} \in H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right) (to be explained below) is non-zero, then H_(f)^(1)(F,rho_(Pi,lambda)(n))H_{f}^{1}\left(F, \rho_{\Pi, \lambda}(n)\right) is generated by Delta_(lambda)\Delta_{\lambda} as an E_(lambda)E_{\lambda}-vector space.
The notion of admissible primes appearing in the theorem consists of a long list of assumptions, some of which are rather technical. Essentially, it guarantees that the Galois
and the reduction mod lambda\lambda representation is suitably large and contains certain particular elements. (This is also related to the notion of VV-general from Theorem 3.3.1.) Fortunately, in some favorable situations, one can show that all but finitely many primes are admissible. For example, this is the case considered in Theorem 3.4.1. For another case in pure automorphic setting, see [49[49, тнм. 1.1.7].
The proof of the theorem uses several different ingredients. The initial step for case (1) is to translate the analytic condition L((1)/(2),Pi)!=0L\left(\frac{1}{2}, \Pi\right) \neq 0 into a more algebraic condition via the global Gan-Gross-Prasad (GGP) conjecture. Namely, the GGP conjecture predicts that in this case, there exist a pair of hermitian spaces (V_(n),V_(n+1))\left(V_{n}, V_{n+1}\right) over FF that are totally positive definite at oo\infty, where V_(n+1)=V_(n)o+FvV_{n+1}=V_{n} \oplus F v with (v,v)=1(v, v)=1, and a tempered cuspidal automorphic representation pi=pi_(n)xxpi_(n+1)\pi=\pi_{n} \times \pi_{n+1} of the product of unitary groups G=U(V_(n))xx U(V_(n+1))G=U\left(V_{n}\right) \times U\left(V_{n+1}\right), such that the period integral
[Delta_(H)]:C_(c)^(**)(Sh(G),E)[pi]rarr E\left[\Delta_{H}\right]: C_{c}^{*}(\operatorname{Sh}(G), E)[\pi] \rightarrow E
does not vanish, where H:=U(V_(n))H:=U\left(V_{n}\right) embeds into GG diagonally, which induces an embedding Delta_(H):Sh(H)↪Sh(G)\Delta_{H}: \operatorname{Sh}(H) \hookrightarrow \operatorname{Sh}(G) of corresponding Shimura varieties (in fact, Shimura sets) with appropriately chosen level structures (here and later we omit level structures from the notations). We denote by [Delta_(H):}\left[\Delta_{H}\right. ] the homology class of Sh(G)\operatorname{Sh}(G) given by Sh(H)\operatorname{Sh}(H) and write C_(c)^(**)(Sh(G),E)[pi]C_{c}^{*}(\operatorname{Sh}(G), E)[\pi] for the pi\pi-isotypical component of the cohomology (i.e., functions) of Sh(G)\operatorname{Sh}(G). This conjecture was first proved in [73] under some local restrictions which are too restrictive for arithmetic applications. Those restrictions are all lifted in our recent work through some new techniques in the study of trace formulae [8].
The strategy then is to construct, for every m >= 1m \geq 1, (infinitely many) cohomology classes {Theta_(m)^(p)}_(p)subH^(1)(F,(rho_(Pi,lambda)//lambda^(m))^(**)(1))\left\{\Theta_{m}^{p}\right\}_{p} \subset H^{1}\left(F,\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1)\right), where pp are appropriately chosen primes and (-)^(**)(1)(-)^{*}(1) denotes the usual Pontryagin duality twisted by the cyclotomic character, such that the local Tate pairing at pp between Theta_(m)^(p)\Theta_{m}^{p} and Selmer classes of the Galois representation rho_(Pi,lambda)//lambda^(m)\rho_{\Pi, \lambda} / \lambda^{m} is related to the above period integral. Then one can use Kolyvagin type argument (amplified in [47,49][47,49] ), with {Theta_(m)^(p)}\left\{\Theta_{m}^{p}\right\} served as annihilators of the Selmer groups, to conclude.
The construction of Theta_(m)^(p)\Theta_{m}^{p} uses the diagonal embedding of Shimura varieties
where H^(')↪G^(')H^{\prime} \hookrightarrow G^{\prime} are prime-to- pp trivialized (extended pure) inner forms of H sub GH \subset G (see Definition 2.3.1). These Shimura varieties have parahoric level structures at pp, and using
8 Here we use the automorphic normalization of the LL-function.
the theory of local models (Section 3.1) one can show that their integral models are polysemistable at pp and compute the sheaf of nearby cycles on their mod pp fibers. Using many ingredients, including the understanding of (integral) cohomology of Sh(G^('))\operatorname{Sh}\left(G^{\prime}\right) over bar(F)\bar{F}, the computations from Example 3.3.3 (in particular, (3.3) and (3.4)), and the Taylor-Wiles patching method [48], one shows that (rho_(Pi,lambda)//lambda^(m))^(**)(1)\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1) does appear in the cohomology of Sh(G^('))\operatorname{Sh}\left(G^{\prime}\right) (the so-called arithmetic level raising for Pi\Pi ), and that the diagonal cycle Delta_(H^('))\Delta_{H^{\prime}}, when localized at (rho_(Pi,lambda)//lambda^(m))^(**)(1)\left(\rho_{\Pi, \lambda} / \lambda^{m}\right)^{*}(1), does give the desired class Theta_(m)^(p)\Theta_{m}^{p}. We shall mention that this is consistent with conjectures in Sections 2.1 and 2.3, as coherent sheaves on Loc_(G,p)oxO_(E)//lambda^(m)\operatorname{Loc}_{G, p} \otimes \mathcal{O}_{E} / \lambda^{m} corresponding to cc-ind K_(p)^(G(Q_(p)))(O_(E)//lambda^(m))K_{p}^{G\left(\mathbb{Q}_{p}\right)}\left(\mathcal{O}_{E} / \lambda^{m}\right) and cc-ind K_(p)^(')G^(')(Q_(p))(O_(E)//lambda^(m))K_{p}^{\prime} G^{\prime}\left(\mathbb{Q}_{p}\right)\left(\mathcal{O}_{E} / \lambda^{m}\right) are expected to be related exactly in this way.
We could also explain the class Delta_(lambda)\Delta_{\lambda} appearing in case (2). Namely, in this case we start with an embedding of Shimura varieties Delta_(H):Sh(H)↪Sh(G)\Delta_{H}: \operatorname{Sh}(H) \hookrightarrow \operatorname{Sh}(G), where GG is a product of unitary groups such that Pi\Pi descends to a tempered cuspidal automorphic representation pi\pi appearing in the middle dimensional cohomology of Sh_(G)\mathrm{Sh}_{G}. Then the pi\pi-isotypical component of the cycle Delta_(H)\Delta_{H} is homologous to zero, and we let Delta_(lambda)=AJ_(lambda)(Delta_(H)[pi])\Delta_{\lambda}=\operatorname{AJ}_{\lambda}\left(\Delta_{H}[\pi]\right). The strategy to prove case (2) then is to reduce it to case (1) via some similar arguments as before.
ACKNOWLEDGMENTS
The author would like to thank all of his collaborators, without whom many works reported in this survey article would not be possible. Besides, he would also like to thank Edward Frenkel, Dennis Gaitsgory, Xuhua He, Michael Rapoport, Peter Scholze for teaching him and for discussions on various parts of the Langlands program over years.
FUNDING
The work is partially supported by NSF under agreement No. DMS-1902239 and a Simons fellowship.
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XINWEN ZHU
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA, xzhu@caltech.edu
4. ALGEBRAIC AND COMPLEX GEOMETRY
MARC LEVINE
ABSTRACT
We give a survey of the development of motivic cohomology, motivic categories, and some of their recent descendants.
Motivic cohomology, motives, KK-theory, algebraic cycles, motivic homotopy theory
1. INTRODUCTION
Motivic cohomology arose out of a marriage of Grothendieck's ideas about motives with a circle of conjectures about special values of zeta functions and LL-functions. It has since taken on a very active life of its own, spawning a multitude of developments and applications. My intention in this survey is to present some of the history of motivic cohomology and the framework that supports it, its current state, and some thoughts about its future directions. I will say very little about the initial impetus given by the conjectures about zeta functions and LL-functions, as there are many others who are much better qualified to tell that story. I will also say next to nothing about the many applications motivic cohomology has seen: I think this would be like writing about the applications of cohomology up to, say, 1950, and would certainly make this already lengthy survey completely unmanageable.
My basic premise is that motivic cohomology is supposed to be universal cohomology for algebro-geometric objects. As "universal" depends on the universe one happens to find oneself in, motivic cohomology is an ever-evolving construct. My plan is to give a path through some of the various universes that have given rise to motivic cohomologies, to describe the resulting motivic cohomologies and put them in a larger, usually categorical, framework. Our path will branch into several directions, reflecting the different aspects of algebraic and arithmetic geometry that have been touched by this theory. We begin with the conjectures of Beilinson and Lichtenbaum about motivic complexes that give rise to the universal Bloch-Ogus cohomology theory on smooth varieties over a field, and the candidate complexes constructed by Bloch and Suslin. We then take up Voevodsky's triangulated category of motives over a field and the embedding of the motivic complexes and motivic cohomology in this framework. The next developments moving further in this direction give us motivic homotopy categories that tell us about "generalized motivic cohomology" for a much wider class of schemes, analogous to the development of the stable homotopy category and generalized cohomology for spaces; this includes a number of candidate theories for motivic cohomology over a general base-scheme. We conclude with three variations on our theme:
Milnor-Witt motives and Milnor-Witt motivic cohomology, incorporating information about quadratic forms,
Motives with modulus, relaxing the usual condition of homotopy invariance with respect to the affine line, and
This last example does not yet, as far as I know, have a categorical framework, while one for a motivic cohomology with modulus is still in development.
There is already an extensive literature on the early development of motives and motivic cohomology. It was not my intention here to cover this part in detail, but I include a section on this topic to give a quick overview for the sake of background, and to isolate a few main ideas so the reader could see how they have influenced later developments.
I would like to thank all those who helped me prepare this survey, especially Tom Bachmann, Federico Binda, Dustin Clausen, Thomas Geisser, Wataru Kai, Akhil Mathew, Hiroyasu Miyazaki, Matthew Morrow, and Shuji Saito. In spite of their efforts, I feel certain that a number of errors have crept in, which are, of course, all my responsibility. I hope that the reader will be able to repair them and continue on.
2. BACKGROUND AND HISTORY
2.1. The conjectures of Beilinson and Lichtenbaum
Beilinson pointed out in his 1983 paper "Higher regulators and values of LL-functions" [13] that the existence of Gillet's Chern character [53] from algebraic KK-theory to an arbitrary Bloch-Ogus cohomology theory [30] with coefficients in a Q\mathbb{Q}-algebra implies that one can form the universal Bloch-Ogus cohomology H_(mu)^(a)(-,Q(b))H_{\mu}^{a}(-, \mathbb{Q}(b)) with Q\mathbb{Q}-coefficients by decomposing algebraic KK-theory into its weight spaces for the Adams operations psi_(k)\psi_{k}. In terms of the indexing, one has
with the Gamma_(X)(b)\Gamma_{X}(b) satisfying a number of axioms. We give Beilinson's list of axioms for motivic complexes in the Zariski topology (axiom iv (p)(p) was added by Milne [90, $2]):
(i) In the derived category of sheaves on X,Gamma(0)X, \Gamma(0) is the constant sheaf Z\mathbb{Z} on Sm_(k)\mathrm{Sm}_{k}, Gamma(1)=G_(m)[-1]\Gamma(1)=\mathbb{G}_{m}[-1] and Gamma(n)=0\Gamma(n)=0 for n < 0n<0.
(ii) The graded object Gamma(**):=[X|->bigoplus_(n >= 0)Gamma_(X)(n)]\Gamma(*):=\left[X \mapsto \bigoplus_{n \geq 0} \Gamma_{X}(n)\right] is a commutative graded ring in the derived category of sheaves on Sm_(k)\mathrm{Sm}_{k}.
(iii) The cohomology sheaves H^(m)(Gamma(n))\mathscr{H}^{m}(\Gamma(n)) are zero for m > nm>n and for m <= 0m \leq 0 if n > 0n>0; H^(n)(Gamma(n))\mathscr{H}^{n}(\Gamma(n)) is the sheaf of Milnor KK-groups X|->K_(n,X)^(M)X \mapsto \mathcal{K}_{n, X}^{M}.
(iv)( pp ) For kk of characteristic p > 0p>0, let W_(v)Omega_("log ")^(n)W_{v} \Omega_{\text {log }}^{n} denote the nu\nu-truncated logarithmic de Rham-Witt sheaf. The d logd \log map d log:K_(n)^(M)//p^(n)rarrW_(nu)Omega_(log)^(n)d \log : \mathcal{K}_{n}^{M} / p^{n} \rightarrow W_{\nu} \Omega_{\log }^{n} induces via (ii) a map Gamma(n)//p^(nu)rarrW_(v)Omega_(log)^(n)[-n]\Gamma(n) / p^{\nu} \rightarrow W_{v} \Omega_{\log }^{n}[-n], which is an isomorphism.
(v) There should also be a spectral sequence starting with integral motivic cohomology and converging to algebraic KK-theory, analogous to the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological KK-theory. Explicitly, this should be
In light of axiom (iii), the Merkurjev-Suslin theorem [89, тHEOREM 14.1] settled the degree >= 2\geq 2 part of (iv)(b) for n=2n=2 even before the complex Gamma\Gamma (2) was defined.
Beilinson [14, $5.10] rephrased and refined these conjectures to a categorical statement, invoking a conjectural category of mixed motivic sheaves, and an embedding of the hypercohomology of the Beilinson-Lichtenbaum complexes into a categorical framework.
In this framework, motivic cohomology should arise via an abelian tensor category of motivic sheaves on Sch_(S),X|->Sh^("mot ")(X)\operatorname{Sch}_{S}, X \mapsto \operatorname{Sh}^{\text {mot }}(X), admitting the six functor formalism of Grothendieck, f^(**),f_(**),f_(!),f^(!),Hom,oxf^{*}, f_{*}, f_{!}, f^{!}, \mathscr{H o m}, \otimes, on the derived categories. There should be Tate objects Z_(X)(n)inSh^("mot ")(X)\mathbb{Z}_{X}(n) \in \operatorname{Sh}^{\text {mot }}(X), and objects M(X):=p_(X!)p_(X)^(!)Z_(S)(0)M(X):=p_{X!} p_{X}^{!} \mathbb{Z}_{S}(0) in the derived category of Sh^(mot)(S),p_(X):X rarr S\operatorname{Sh}^{\mathrm{mot}}(S), p_{X}: X \rightarrow S the structure morphism, and motivic cohomology should arise as the Hom-groups
2.2. Bloch's higher Chow groups and Suslin homology
The first good definition of motivic cohomology complexes was given by Spencer Bloch, in his landmark 1985 paper "Algebraic cycles and higher Chow groups" [24]. As
cycles modulo rational equivalence.
For XX a finite type kk-scheme, recall that the group of dimension dd algebraic cycles on X,Z_(d)(X)X, \mathrm{Z}_{d}(X), is the free abelian group on the integral closed subschemes ZZ of XX of dimension dd over kk. The group of cycles modulo rational equivalence, CH_(d)(X)\mathrm{CH}_{d}(X), has the following presentation. Let n|->Delta^(n)n \mapsto \Delta^{n} be the cosimplicial scheme of algebraic nn-simplices
The coface and codegeneracy maps are defined just as for the usual real simplices Delta_("top ")^(n)subR^(n)\Delta_{\text {top }}^{n} \subset \mathbb{R}^{n}. A face of Delta^(n)\Delta^{n} is a closed subscheme defined by the vanishing of some of the t_(i)t_{i}. Let z_(d)(X,n)z_{d}(X, n) be the subgroup of the (n+d)(n+d)-dimensional algebraic cycles Z_(n+d)(X xxDelta^(n))Z_{n+d}\left(X \times \Delta^{n}\right) generated by the integral closed W sub X xxDelta^(n)W \subset X \times \Delta^{n} such that dim W nn X xx F=m+d\operatorname{dim} W \cap X \times F=m+d for each mm-dimensional face FF (or the intersection is empty). For cycles w inz_(d)(X,n)w \in z_{d}(X, n), the face condition gives a well-defined pullback (Id_(X)xx g)^(**):z_(d)(X,n)rarrz_(d)(X,m)\left(\operatorname{Id}_{X} \times g\right)^{*}: z_{d}(X, n) \rightarrow z_{d}(X, m) for each map g:Delta^(m)rarrDelta^(n)g: \Delta^{m} \rightarrow \Delta^{n} in the cosimplicial structure, forming the simplicial abelian group n|->z_(d)(X,n)n \mapsto z_{d}(X, n) and giving the associated chain complex z_(d)(X,**)z_{d}(X, *), Bloch's cycle complex. The degree 0 and 1 terms of z_(d)(X,**)z_{d}(X, *) give our promised presentation of CH_(d)(X)\mathrm{CH}_{d}(X),
With some technical difficulties due to the necessity of invoking moving lemmas to allow for pullback morphisms, the assignment
X|->z^(q)(X,2q-**)X \mapsto z^{q}(X, 2 q-*)
can be modified via isomorphisms in the derived category to a presheaf of cohomological complexes Z_(Bl)(q)\mathbb{Z}_{B l}(q) on Sm_(k)\mathrm{Sm}_{k}.
After Bloch introduced his cycle complexes, Suslin [111] constructed an algebraic version of singular homology. For a kk-scheme XX, instead of a naive generalization of the singular chain complex of a topological space by taking the free abelian group on the morphisms Delta_(k)^(n)rarr X\Delta_{k}^{n} \rightarrow X, Suslin's insight was to enlarge this to the abelian group of finite correspondences.
A subvariety WW of a product Y xx XY \times X of varieties (with YY smooth) defines an irreducible finite correspondence from YY to XX if p_(1):W rarr Yp_{1}: W \rightarrow Y is finite and surjective to some irreducible component of YY. The association y|->p_(2)(p_(1)^(-1)(y))y \mapsto p_{2}\left(p_{1}^{-1}(y)\right) can be thought of as a multivalued map from YY to XX.
The group of finite correspondences Cor_(k)(Y,X)\operatorname{Cor}_{k}(Y, X) is defined as the free abelian group on the irreducible finite correspondences. Given a morphism f:Y^(')rarr Yf: Y^{\prime} \rightarrow Y, there is a pullback map f^(**):Cor_(k)(Y,X)rarrCor_(k)(Y^('),X)f^{*}: \operatorname{Cor}_{k}(Y, X) \rightarrow \operatorname{Cor}_{k}\left(Y^{\prime}, X\right), compatible with the interpretation as multivalued functions, and making Cor_(k)(-,X)\operatorname{Cor}_{k}(-, X) into a contravariant functor from smooth varieties over kk to abelian groups.
Suslin defines C_(n)^("Sus ")(X):=Cor_(k)(Delta_(k)^(n),X)C_{n}^{\text {Sus }}(X):=\operatorname{Cor}_{k}\left(\Delta_{k}^{n}, X\right); the structure of Delta_(k)^(**)\Delta_{k}^{*} as smooth cosimplicial scheme makes n|->C_(n)^("Sus ")(X)n \mapsto C_{n}^{\text {Sus }}(X) a simplicial abelian group. As above, we have the associated complex C_(**)^("Sus ")(X)C_{*}^{\text {Sus }}(X), the Suslin complex of XX, whose homology is the Suslin homology of XX :
In fact, the monoid of the N\mathbb{N}-linear combinations of irreducible correspondences W sub X xx YW \subset X \times Y is the same as the monoid of morphisms
phi:X rarr⨆_(n >= 0)Sym^(n)Y\phi: X \rightarrow \bigsqcup_{n \geq 0} \operatorname{Sym}^{n} Y
where Sym^(n)Y\operatorname{Sym}^{n} Y is the quotient Y^(n)//Sigma_(n)Y^{n} / \Sigma_{n} of Y^(n)Y^{n} by the symmetric group permuting the factors, with the monoid structure induced by the sum map
Sym^(n)Y xxSym^(m)Y rarrSym^(m+n)Y\operatorname{Sym}^{n} Y \times \operatorname{Sym}^{m} Y \rightarrow \operatorname{Sym}^{m+n} Y
Suslin's complex and his definition of algebraic homology can thus be thought of as an algebraic incarnation of the theorem of Dold-Thom [34, sATz 6.4], that identifies the homotopy groups of the infinite symmetric product of a pointed CW\mathrm{CW} complex TT with the
reduced homology of TT. The main result of [112] gives an isomorphism of the mod nn Suslin homology, H_(**)^("Sus ")(X,Z//n)H_{*}^{\text {Sus }}(X, \mathbb{Z} / n), for XX of finite type over C\mathbb{C}, with the mod n\bmod n singular homology of X(C)X(\mathbb{C}), a first major success of the theory.
Let Delta_("top ")^(n)\Delta_{\text {top }}^{n} denote the usual nn-simplex
2.4. Voevodsky's category DM and modern motivic cohomology
One can almost realize Beilinson's ideas of a categorical framework for motivic cohomology by working in the setting of triangulated categories, viewed as a replacement for the derived category of Beilinson's conjectured abelian category of motivic sheaves. Once this is accomplished, one could hope that an abelian category of mixed motives could be constructed out of the triangulated category as the heart of a suitable tt-structure.
Constructions of a triangulated category of mixed motives over a perfect base-field were given by Hanamura [57-59], Voevodsky [127], and myself [83]. All three categories yield Bloch's higher Chow groups as the categorical motivic cohomology, however, Voevodsky's sheaf-theoretic approach has had the most far-reaching consequences and has been widely adopted as the correct solution. The construction of a motivic tt-structure is still an open problem. ^(1){ }^{1} There are also constructions of triangulated categories of mixed motives by the method of compatible realizations, such as by Huber [64], or Nori's construction of an abelian category of mixed motives, described in [65, PART II]; we will not pursue these directions here. We also refer the reader to Jannsen's survey on mixed motives [68].
Voevodsky's triangulated category of motives over k,DM(k)k, \operatorname{DM}(k), is based on the category of finite correspondences on Sm_(k)\mathrm{Sm}_{k}, a refinement of Grothendieck's composition law for correspondences on smooth projective varieties. Grothendieck had constructed categories of motives for smooth projective varieties, with the morphisms from XX to YY given by the group of cycles modulo rational equivalence CH_(dim X)(X xx Y)\mathrm{CH}_{\operatorname{dim} X}(X \times Y). The composition law is given by
with W inCH_(dim X)(X xx Y)W \in \mathrm{CH}_{\operatorname{dim} X}(X \times Y) and W^(')inCH_(dim Y)(Y xx Z)W^{\prime} \in \mathrm{CH}_{\operatorname{dim} Y}(Y \times Z); one needs to pass to cycle classes to define p_(XY)^(**)(W)*p_(YZ)^(**)(W^('))p_{X Y}^{*}(W) \cdot p_{Y Z}^{*}\left(W^{\prime}\right) and the projection p_(XZ)p_{X Z} needs to be proper (that is, YY needs to be proper over kk ) to define p_(XZ**)p_{X Z *}.
Voevodsky's key insight was to restrict to finite correspondences, so that all the operations used in the composition law of correspondence classes would be defined on the level of the cycles themselves, without needing to pass to rational equivalence classes, and without needing the varieties involved to be proper. Voevodsky's idea of having a well-defined composition law on a restricted class of correspondences has been modified and applied in a wide range of different contexts, somewhat similar to the use of various flavors of bordism theories in topology.
Let XX and YY be in Sm_(k)\operatorname{Sm}_{k}. Recall from Section 2.2 the subgroup Cor_(k)(X,Y)sub\operatorname{Cor}_{k}(X, Y) \subsetZ_(dim X)(X xx Y)Z_{\operatorname{dim} X}(X \times Y) generated by the integral W sub X xx YW \subset X \times Y that are finite over XX and map surjectively to a component of XX.
Lemma 2.3. Let X,Y,ZX, Y, Z be smooth kk-varieties and take alpha inCor_(k)(X,Y),beta inCor_(k)(Y,Z)\alpha \in \operatorname{Cor}_{k}(X, Y), \beta \in \operatorname{Cor}_{k}(Y, Z). Then
(i) The cycles p_(YZ)^(**)(beta)p_{Y Z}^{*}(\beta) and p_(XY)^(**)(alpha)p_{X Y}^{*}(\alpha) intersect properly on X xx Y xx ZX \times Y \times Z, so the inter section product p_(YZ)^(**)(beta)*p_(XY)^(**)(alpha)p_{Y Z}^{*}(\beta) \cdot p_{X Y}^{*}(\alpha) exists as a well-defined cycle on X xx Y xx ZX \times Y \times Z.
(ii) Letting |alpha|sub X xx Y|\alpha| \subset X \times Y, and |beta|sub Y xx Z|\beta| \subset Y \times Z denote the support of alpha\alpha and beta\beta, respectively, each irreducible component of the intersection X xx|beta|nn|alpha|xx ZX \times|\beta| \cap|\alpha| \times Z is finite over X xx ZX \times Z, and maps surjectively onto some component of XX.
makes sense for alpha inCor_(k)(X,Y)\alpha \in \operatorname{Cor}_{k}(X, Y) and beta inCor_(k)(Y,Z)\beta \in \operatorname{Cor}_{k}(Y, Z), and the resulting cycle on X xx ZX \times Z is in Cor_(k)(X,Z)\operatorname{Cor}_{k}(X, Z). This defines the composition law in Voevodsky's category of finite correspondences, Cor_(k)\operatorname{Cor}_{k}, with objects as for Sm_(k)\operatorname{Sm}_{k}, and morphisms Hom_(Cor_(k))(X,Y)=Cor_(k)(X,Y)\operatorname{Hom}_{\operatorname{Cor}_{k}}(X, Y)=\operatorname{Cor}_{k}(X, Y). Sending a usual morphism f:X rarr Yf: X \rightarrow Y of smooth varieties to its graph defines a faithful functor [-]:Sm_(k)rarr[-]: \mathrm{Sm}_{k} \rightarrow Cor _(k)_{k}.
Once one has the category Cor_(k)\mathrm{Cor}_{k}, the path to DM(k)\mathrm{DM}(k) is easy to describe. One takes the category of additive presheaves of abelian groups on Cor_(k)\mathrm{Cor}_{k}, the category of presheaves with transfer PST(k)\operatorname{PST}(k). Inside PST(k)\operatorname{PST}(k) is the category NST (k)(k) of Nisnevich sheaves with transfer, that is, a presheaf that is a Nisnevich sheaf when restricted to Sm_(k)subCor_(k)\operatorname{Sm}_{k} \subset \operatorname{Cor}_{k}. Each X inSm_(k)X \in \operatorname{Sm}_{k} defines an object Z_(tr)(X)in NST(k)\mathbb{Z}_{\mathrm{tr}}(X) \in \operatorname{NST}(k), as the representable (pre)sheaf Y|->Cor_(k)(Y,X)Y \mapsto \operatorname{Cor}_{k}(Y, X). Inside the derived category D(NST(k))D(\operatorname{NST}(k)) is the full subcategory of complexes KK whose homology presheaves h__(i)(K)\underline{h}_{i}(K) are A^(1)\mathbb{A}^{1}-homotopy invariant: h__(i)(K)(X)~=h__(i)(K)(X xxA^(1))\underline{h}_{i}(K)(X) \cong \underline{h}_{i}(K)\left(X \times \mathbb{A}^{1}\right) for all X inSm_(k)X \in \mathrm{Sm}_{k}. This is the category of effective motives DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k). The Suslin complex construction, P|->C_(**)^("Sus ")(P)\mathcal{P} \mapsto C_{*}^{\text {Sus }}(\mathcal{P}), with
extends to a functor RC_(**)^("Sus "):D(NST(k))rarrDM^("eff ")(k)R C_{*}^{\text {Sus }}: D(\mathrm{NST}(k)) \rightarrow \mathrm{DM}^{\text {eff }}(k), and realizes DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k) as the localization of D(NST(k))D(\operatorname{NST}(k)) with respect to the complexes Z_(tr)(X xxA^(1))rarr"p_(**)"Z_(tr)(X)\mathbb{Z}_{\mathrm{tr}}\left(X \times \mathbb{A}^{1}\right) \xrightarrow{p_{*}} \mathbb{Z}_{\mathrm{tr}}(X). Via RC_(**)^("Sus ")R C_{*}^{\text {Sus }}DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k) inherits a tensor structure ox\otimes from D(NST(k))D(\mathrm{NST}(k)). The functor Z_(tr):Sm_(k)rarrNST(k)\mathbb{Z}_{\mathrm{tr}}: \mathrm{Sm}_{k} \rightarrow \mathrm{NST}(k) defines the functor M^("eff "):=RC_(**)^(Sus)@Z_(tr)M^{\text {eff }}:=R C_{*}^{\mathrm{Sus}} \circ \mathbb{Z}_{\mathrm{tr}},
The Tate object Z(1)inDM^(eff)(k)\mathbb{Z}(1) \in \mathrm{DM}^{\mathrm{eff}}(k) is the image of the complex Z_(tr)(\mathbb{Z}_{\mathrm{tr}}( Spec k)rarr"i_(oo**)"k) \xrightarrow{i_{\infty *}}Z_(tr)(P^(1))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{P}^{1}\right) (with Z_(tr)(P^(1))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{P}^{1}\right) in degree 2 ) via RC_(**)^("Sus ")R C_{*}^{\text {Sus }}. One forms the triangulated tensor category DM(k)\operatorname{DM}(k) as the category of -oxZ(1)-\otimes \mathbb{Z}(1)-spectrum objects in DM^("eff ")(k)\mathrm{DM}^{\text {eff }}(k), inverting the endofunctor -oxZ(1)-\otimes \mathbb{Z}(1); for M inDM(k)M \in \mathrm{DM}(k), one has the Tate twists M(n):=M ox Z(1)^(ox n)M(n):=M \otimes Z(1)^{\otimes n} for n inZn \in \mathbb{Z}; in particular, we have the Tate objects Z(n)\mathbb{Z}(n). The functor M^("eff ")M^{\text {eff }} induces the functor M:Sm_(k)rarrDM(k)M: \mathrm{Sm}_{k} \rightarrow \mathrm{DM}(k).
Bloch's higher Chow groups, Suslin homology, and the motivic complexes Z_(BI)(q)\mathbb{Z}_{\mathrm{BI}}(q) are represented in DM(k)\mathrm{DM}(k) via canonical isomorphisms
where Z_(tr)(G_(m))\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{G}_{m}\right) is the quotient presheaf Z_(tr)(A^(1)\\{0})//Z_(tr)({1})\mathbb{Z}_{\mathrm{tr}}\left(\mathbb{A}^{1} \backslash\{0\}\right) / \mathbb{Z}_{\mathrm{tr}}(\{1\}). The complexes Z_(V)(q)\mathbb{Z}_{V}(q) and Z_(BI)(q)\mathbb{Z}_{\mathrm{BI}}(q) define isomorphic objects in DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k), in particular, are isomorphic in the derived category of Nisnevich sheaves on Sm_(k)\mathrm{Sm}_{k}. The details of these constructions and results are carried out in [127] (with a bit of help from [117]).
2.5. Motivic homotopy theory
Although Voevodsky's triangulated category of motives does give motivic cohomology a categorical foundation, this is really a halfway station on the way to a really suitable categorical framework. As analogy, embedding the Beilinson-Lichtenbaum/Bloch-Suslin theory of motivic complexes in DM(k)\mathrm{DM}(k) is like considering the singular chain or cochain complex of a topological space as an object in the derived category of abelian groups. A much more fruitful framework for singular (co)homology is to be found in the stable homotopy category SH.
A parallel representability for motivic cohomology for schemes over a base-scheme BB in a wider category of good cohomology theories is to be found in the motivic stable homotopy category over B,SH(B)B, \mathrm{SH}(B). This, together with the motivic unstable homotopy category, H(B)\mathscr{H}(B), gives the proper setting for the deeper study of motivic cohomology, besides placing this theory on a equal footing with all cohomology theories on algebraic varieties that satisfy a few natural axioms.
In topology, the representation of singular (co)homology via the singular (co)chain complexes is placed in the setting of stable homotopy theory through the construction of the
Eilenberg-MacLane spectra, giving a natural isomorphism for each abelian group AA,
H^(n)(X,A)~=Hom_(SH)(Sigma^(oo)X_(+),Sigma^(n)EM(A))H^{n}(X, A) \cong \operatorname{Hom}_{\mathrm{SH}}\left(\Sigma^{\infty} X_{+}, \Sigma^{n} \operatorname{EM}(A)\right)
with the Eilenberg-MacLane spectrum EM(A)\operatorname{EM}(A) being characterized by its stable homotopy groups
pi_(n)^(s)(EM(A))={[A," for "n=0],[0," else "]:}\pi_{n}^{s}(\operatorname{EM}(A))= \begin{cases}A & \text { for } n=0 \\ 0 & \text { else }\end{cases}
The assignment A|->EM(A)A \mapsto \operatorname{EM}(A) extends to a fully faithful embedding EM : D(Ab)rarrSHD(\mathbf{A b}) \rightarrow \mathrm{SH}. This realizes the ordinary (co)homology as being represented by the derived category D(Ab)D(\mathbf{A b}) via its Eilenberg-MacLane embedding in SH, which in turn is to be viewed as the category of all cohomology theories on reasonable topological spaces.
The stable homotopy category SH is the stabilization of the unstable pointed homotopy category H_(∙)\mathscr{H}_{\bullet} with respect to the suspension operator Sigma X:=S^(1)^^X\Sigma X:=S^{1} \wedge X, which becomes an invertible endofunctor on SH\mathrm{SH}. The resulting functor of H_(∙)\mathscr{H}_{\bullet} to its stabilization is the infinite suspension functor Sigma^(oo)\Sigma^{\infty} and gives us the "effective" subcategory SH^("eff ")subSH\mathrm{SH}^{\text {eff }} \subset \mathrm{SH}, as the smallest subcategory containing Sigma^(oo)(H_(∙))\Sigma^{\infty}\left(\mathscr{H}_{\bullet}\right) and closed under homotopy cofibers and small coproducts. This in turn gives a decreasing filtration on SH\mathrm{SH} by the subcategories Sigma^(n)SH^(eff)\Sigma^{n} \mathrm{SH}^{\mathrm{eff}}, n inZn \in \mathbb{Z}. This rather abstract looking filtration is simply the filtration by connectivity: EE is in Sigma^(n)SH^(eff)\Sigma^{n} \mathrm{SH}^{\mathrm{eff}} if and only if pi_(m)^(s)E=0\pi_{m}^{s} E=0 for m < nm<n. The layers in this filtration are isomorphic to the category Ab\mathbf{A b}, by the functor E|->pi_(n)EE \mapsto \pi_{n} E, and in fact, this filtration is the one given by a natural tt-structure on SH\mathrm{SH} with heart Ab\mathbf{A b}; concretely, the 0 th truncation tau_(0)E\tau_{0} E is given by the Eilenberg-MacLane spectrum EM(pi_(0)(E))\operatorname{EM}\left(\pi_{0}(E)\right).
A central example is the sphere spectrum S:=Sigma^(oo)S^(0)\mathbb{S}:=\Sigma^{\infty} S^{0}. Since
we have tau_(0)S=EM(Z)\tau_{0} \mathbb{S}=\operatorname{EM}(\mathbb{Z}), establishing the natural relation between homology and homotopy.
In the motivic world, we have a somewhat parallel picture. The pointed unstable category H_(∙)(B)H_{\bullet}(B) has a natural 2-parameter family of "spheres." Let S^(n)S^{n} denote the constant presheaf with value the pointed nn-sphere, and let G_(m)\mathbb{G}_{m} denote the representable presheaf A^(1)\\{0}\mathbb{A}^{1} \backslash\{0\} pointed by 1 . Define
for a >= b >= 0a \geq b \geq 0. We consider P^(1)\mathbb{P}^{1} as the representable presheaf, pointed by 1 ; there is a canonical isomorphism P^(1)~=S^(2,1)\mathbb{P}^{1} \cong S^{2,1} in H_(∙)(B)\mathscr{H}_{\bullet}(B).
In order to achieve the analog of Spanier-Whitehead duality in the motivic setting, one needs to use spectra with respect to P^(1)\mathbb{P}^{1}-suspension rather than with respect to S^(1)S^{1}-suspension. The category SH(B)\mathrm{SH}(B) is constructed as a homotopy category of P^(1)\mathbb{P}^{1}-spectra in H_(∙)(B)\mathscr{H}_{\bullet}(B), so P^(1)\mathbb{P}^{1}-suspension becomes invertible and our family of spheres extends to a family of invertible suspension endofunctors
Sigma^(a,b):SH(B)rarrSH(B),quad a,b inZ\Sigma^{a, b}: \mathrm{SH}(B) \rightarrow \mathrm{SH}(B), \quad a, b \in \mathbb{Z}
Each E inSH(B)E \in \mathrm{SH}(B) gives the bigraded cohomology theory on Sm_(B)\mathrm{Sm}_{B} by
Note that the translation in SH(B)\mathrm{SH}(B) is given by S^(1)S^{1}-suspension, not P^(1)\mathbb{P}^{1}-suspension.
The effective subcategory SH^("eff ")(B)\mathrm{SH}^{\text {eff }}(B) is defined as the localizing subcategory (i.e., a triangulated subcategory closed under small coproducts) generated by the P^(1)\mathbb{P}^{1}-infinite suspension spectra Sigma_(P^(1))^(oo)X\Sigma_{\mathbb{P}^{1}}^{\infty} \mathcal{X} for XinH_(∙)(B)\mathcal{X} \in \mathscr{H}_{\bullet}(B). We replace the filtration of SH\mathrm{SH} with respect to S^(1)S^{1}-connectivity with the filtration on SH(B)\mathrm{SH}(B) with respect to P^(1)\mathbb{P}^{1}-connectivity, via the subcategories Sigma_(P_(1)^(1))^(n)SH^(eff)(B)\Sigma_{\mathbb{P}_{1}^{1}}^{n} \mathrm{SH}^{\mathrm{eff}}(B). This is Voevodsky's slice filtration, with associated nnth truncation denoted f_(n)f_{n}, giving for each E inSH(B)E \in \mathrm{SH}(B) the tower
cdots rarrf_(n+1)E rarrf_(n)E rarr cdots rarr E\cdots \rightarrow f_{n+1} E \rightarrow f_{n} E \rightarrow \cdots \rightarrow E
One has the layers s_(n)Es_{n} E of this tower, fitting into a distinguished triangle
f_(n+1)E rarrf_(n)E rarrs_(n)E rarrf_(n+1)E[1]=Sigma^(1,0)f_(n+1)Ef_{n+1} E \rightarrow f_{n} E \rightarrow s_{n} E \rightarrow f_{n+1} E[1]=\Sigma^{1,0} f_{n+1} E
An important difference from the topological case is that this is a filtration by triangulated subcategories; the P^(1)\mathbb{P}^{1}-suspension is not the shift in the triangulated structure on SH(B)\mathrm{SH}(B), and so the slice filtration does not arise from a tt-structure.
We concentrate for a while on the case B=Spec k,kB=\operatorname{Spec} k, k a perfect field. There is an Eilenberg-MacLane functor
see also [85, THEOREM 10.5.1] and the recent paper of Bachmann-Elmanto [9]. In other words, the 0th slice truncation of the motivic sphere spectrum represents motivic cohomology. Röndigs-Østvær [103] show that the homotopy category of EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))-modules in SH(k)\mathrm{SH}(k) is equivalent to DM(k)\mathrm{DM}(k) and represents the Eilenberg-MacLane functor as the forgetful functor, right-adjoint to the free EM(Z(0))\operatorname{EM}(\mathbb{Z}(0)) functor
DM(k)\operatorname{DM}(k)
This is the triangulated motivic analog of the classical result, that the heart of the tt-structure on SH\mathrm{SH} is Ab\mathbf{A b}.
2.6. Motivic cohomology and the rational motivic stable homotopy category
In classical homotopy theory, the Eilenberg-MacLane functor EM : D(Ab)rarrSHD(\mathbf{A b}) \rightarrow \mathrm{SH} has a nice structural property: after Q\mathbb{Q}-localization, the functor EM_(Q):D(Ab)_(Q)rarrSH_(Q)\mathrm{EM}_{\mathbb{Q}}: D(\mathbf{A b})_{\mathbb{Q}} \rightarrow \mathrm{SH}_{\mathbb{Q}} is an equivalence. Does the same happen for the motivic Eilenberg-MacLane functor EM:DM(k)rarrSH(k)\mathrm{EM}: \mathrm{DM}(k) \rightarrow \mathrm{SH}(k) ? In general, the answer is no, and the reason goes back to Morel's C-R\mathbb{C}-\mathbb{R} dichotomy for SH(k)\mathrm{SH}(k).
We discuss the case of a characteristic zero field kk as base. Suppose that kk admits a real embedding sigma:k rarrR\sigma: k \rightarrow \mathbb{R}. The embedding sigma\sigma induces a realization functor
which sends the P^(1)\mathbb{P}^{1}-suspension spectrum Sigma_(P^(1))^(oo)X_(+)\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}of a smooth kk-scheme XX to the infinite suspension spectrum of the real manifold of real points X(R)X(\mathbb{R}). For an embedding sigma:k rarrC\sigma: k \rightarrow \mathbb{C}, one has the realization functor R_(C)^(sigma):SH(k)rarrSH\mathfrak{R}_{\mathbb{C}}^{\sigma}: \mathrm{SH}(k) \rightarrow \mathrm{SH}, sending Sigma_(P^(1))^(oo)X_(+)\Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}to Sigma^(oo)X(C)_(+)\Sigma^{\infty} X(\mathbb{C})_{+}. If we take X=P^(1)X=\mathbb{P}^{1}, the real embedding gives you SigmaS\Sigma \mathbb{S} and the complex embedding yields Sigma^(2)S\Sigma^{2} \mathbb{S}, since P^(1)(R)=S^(1),P^(1)(C)=S^(2)\mathbb{P}^{1}(\mathbb{R})=S^{1}, \mathbb{P}^{1}(\mathbb{C})=S^{2}. This has the effect that the switch map tau:P^(1)^^P^(1)rarrP^(1)^^P^(1)\tau: \mathbb{P}^{1} \wedge \mathbb{P}^{1} \rightarrow \mathbb{P}^{1} \wedge \mathbb{P}^{1} induces an automorphism of S_(k)\mathbb{S}_{k} that maps to -1 under the real embedding and to +1 under the complex embedding. Thus, if we invert 2 and decompose the motivic sphere spectrum into +-1\pm 1 eigenfactors with respect to tau\tau, we decompose SH(k)[1//2]\mathrm{SH}(k)[1 / 2] into corresponding summands SH(k)_(+-)\mathrm{SH}(k)_{ \pm}, with all of SH(k)_(+)\mathrm{SH}(k)_{+}going to zero under the real embedding and all of
Alternatively, the minus part is SH(k)[1//2,eta^(-1)]\operatorname{SH}(k)\left[1 / 2, \eta^{-1}\right], where eta\eta is the P^(1)\mathbb{P}^{1}-stabilization of the algebraic Hopf map
A motivic spectrum E inSH(k)E \in \mathrm{SH}(k) is orientable if EE has a good theory of Thom classes. For V rarr XV \rightarrow X a vector bundle with 0 -section s_(0):X rarr Vs_{0}: X \rightarrow V, we have the Thom space Th(V):=V//(V\\s_(0)(X))inH_(∙)(k)\operatorname{Th}(V):=V /\left(V \backslash s_{0}(X)\right) \in \mathscr{H}_{\bullet}(k) (defined as the quotient of representable presheaves). An orientation for EE consists of giving a class
for each rank rr vector bundle V rarr XV \rightarrow X over X inSm_(k)X \in \mathrm{Sm}_{k}, satisfying axioms parallel to the notion of a C\mathbb{C}-orientation in topology; a choice of Thom classes defines EE as an oriented cohomology theory. After inverting 2, all the orientable EE live in the plus part; this includes motivic cohomology, as well as algebraic KK-theory and algebraic cobordism. These theories EE all have the property that eta\eta induces zero on EE-cohomology.
Theories that live in the minus part will contrariwise invert eta\eta (after inverting 2); these include things like Witt theory or cohomology of the sheaf of Witt groups. The real and complex avatars of this are seen by noting that the complex realization of the algebraic Hopf map is the usual Hopf map, which is the 2-torsion element of stable pi_(1)\pi_{1} of the sphere spectrum, while the real realization is the multiplication map xx2:S^(1)rarrS^(1)\times 2: S^{1} \rightarrow S^{1}.
The rational minus part is also a homotopy category of modules over a suitable cohomology theory, namely Witt sheaf cohomology. For a field FF, we have the Witt ring W(F)W(F) of virtual non-degenerate quadratic forms, modulo the hyperbolic form. This extends to a sheaf W\mathcal{W} on Sm_(k)\mathrm{Sm}_{k}, and the functor X|->H_(Nis)^(p)(X,W)X \mapsto H_{\mathrm{Nis}}^{p}(X, \mathcal{W}) is represented in SH(k)\mathrm{SH}(k) by a suitable spectrum EM(W)\operatorname{EM}(\mathcal{W}). We have
Theorem 2.5 ([3, THEOREM 4.2, COROLLARY 4.4]). The functor E|->EM(W)_(Q)^^EE \mapsto \mathrm{EM}(\mathcal{W})_{\mathbb{Q}} \wedge E induces a natural isomorphism of SH(k)_(-Q)\mathrm{SH}(k)_{-\mathbb{Q}} with the homotopy category EM(W)_(Q)\mathrm{EM}(\mathcal{W})_{\mathbb{Q}}-modules.
From this point of view, one can see the Z\mathbb{Z}-graded cohomology theory
as the motivic cohomology for the minus part; this theory picks up information about the real points of schemes. To get the complete theory, one also needs to include twists of W\mathcal{W} by line bundles, an analog of orientation local systems in the topological setting. We will say more about this in Section 4.
2.7. Slice tower and motivic Atiyah-Hirzebruch spectral sequences
The classical Atiyah-Hirzebruch spectral sequence for a spectrum E inSHE \in \mathrm{SH} is the spectral sequence of the Postnikov tower of EE, and looks like
This comes by identifying the qq th layer in the Postnikov tower with the shifted EilenbergMacLane spectrum Sigma^(q)EM(pi_(q)(E))\Sigma^{q} \mathrm{EM}\left(\pi_{q}(E)\right).
Together with results of Pelaez [99] and Gutierrez-Röndigs-Spitzweck, Voevodsky's isomorphism (2.2) has a structural expression, namely, for any E inSH(k)E \in \mathrm{SH}(k), each slice s_(q)(E)s_{q}(E) has a canonical structure of an EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))-module. We write corresponding object of DM(k)\operatorname{DM}(k) as pi_(q)^(mot)(E)\pi_{q}^{\operatorname{mot}}(E), satisfying
s_(q)(E)=Sigma_(P^(1))^(q)EM(pi_(q)^(mot)(E))=S^(2q,q)^^EM(pi_(q)^(mot)(E))s_{q}(E)=\Sigma_{\mathbb{P}^{1}}^{q} \operatorname{EM}\left(\pi_{q}^{\operatorname{mot}}(E)\right)=S^{2 q, q} \wedge \operatorname{EM}\left(\pi_{q}^{\operatorname{mot}}(E)\right)
This gives the motivic Atiyah-Hirzebruch spectral sequence
These slices have been explicitly identified in a number of important cases. The first case was algebraic KK-theory, KGLinSH(k)\mathrm{KGL} \in \mathrm{SH}(k). Voevodsky [118,119] and Levine [85] show
corresponding to classical computation for topological KK-theory,
pi_(q)^(s)KU={[Z," for "q" even "],[0," for "q" odd. "]:}\pi_{q}^{s} K U= \begin{cases}\mathbb{Z} & \text { for } q \text { even } \\ 0 & \text { for } q \text { odd. }\end{cases}
Using "algebraic Bott periodicity" for KGL: KGL^(a+2n,b+n)(X)=KGL^(a,b)(X)=K_(2b-a)(X)\mathrm{KGL}^{a+2 n, b+n}(X)=\mathrm{KGL}^{a, b}(X)=K_{2 b-a}(X), this yields the Atiyah-Hirzebruch spectral sequence of the Beilinson-Lichtenbaum axiom (v),
There is also a corresponding spectral sequence with Z//m\mathbb{Z} / m-coefficients.
This Atiyah-Hirzebruch spectral sequence for algebraic KK-theory was first constructed for XX the spectrum of a field by Bloch and Lichtenbaum [29], by a completely different approach and without recourse to motivic homotopy theory or Voevodsky's slice tower. Their construction was generalized to general X inSm_(k)X \in \mathrm{Sm}_{k} by Friedlander-Suslin [43], also without using the categorical machinery. The rough idea is to give a filtration by codimension of support on X xxDelta^(**)X \times \Delta^{*} (with additional conditions), and then identify the layers with a suitable complex of cycles. Another approach, by Grayson [54], relies on the KK-theory of exact categories with commuting isomorphisms. For smooth finite-type schemes over a perfect field, all these approaches yield the same spectral sequence (see [85, THEOREM 7.1.1, THEOREM 9.0.3], [44])).
3. MOTIVIC COHOMOLOGY OVER A GENERAL BASE
It is natural to ask if this picture of a good motivic cohomology theory for schemes over a perfect field can be extended to more general base-schemes, not just as an interesting technical question but for a wide range of applications, especially in arithmetic. Over a perfect field, we have a number of different constructions that all lead to the same groups, each of which have their advantages and disadvantages: Bloch's higher Chow groups, the cohomology of a suitable Suslin complex, the morphisms in DM(k)\mathrm{DM}(k), the cohomology theory represented in SH(k)\mathrm{SH}(k) by EM(Z(0))\operatorname{EM}(\mathbb{Z}(0)), or by s_(0)S_(k)s_{0} \mathbb{S}_{k}, or by s_(0)KGLs_{0} \mathrm{KGL}.
One would expect motivic cohomology to be an absolute theory, like algebraic K-K- theory, that is, its value on a given scheme should not depend on the choice of base-scheme. In terms of a spectrum HZ_(S)inSH(S)H \mathbb{Z}_{S} \in \mathrm{SH}(S) that would represent our putative theory, this is the cartesian condition: there should be canonical isomorphisms HZ_(T)~=f^(**)HZ_(S)H \mathbb{Z}_{T} \cong f^{*} H \mathbb{Z}_{S} for each morphism of schemes f:T rarr Sf: T \rightarrow S.
The identity (2.2) raises the possibility of defining motivic cohomology over a general base-scheme BB by this formula. One problem here is that the slice filtration has only a limited functoriality: for f:C rarr Bf: C \rightarrow B a map of schemes, one does not in general have a natural isomorphism f^(**)@s_(0)~=s_(0)@f^(**)f^{*} \circ s_{0} \cong s_{0} \circ f^{*}. For the cartesian property to hold for a motivic cohomology defined via the slice filtration, one would want the compatibility of the slices
with pullback; this latter is in fact the case for f:C rarr Bf: C \rightarrow B is a morphism of separated, finite type schemes over a field kk of characteristic zero (or assuming resolution of singularities for separated, finite type kk-schemes), by results of Pelaez [100, COROLLARY 4.3]. This compatibility also holds for arbitrary smooth ff, but is not known in general.
Another concrete candidate for the motivic Borel-Moore homology is given by the hypercohomology of a version of Bloch's cycle complex, suitably extended to the setting of finite type schemes over a Dedekind domain. This theory is nearly absolute, as it depends only on a good notion of dimension or codimension, which one would have for say equiKrull-dimensional schemes. In general, however, this theory lacks a full functoriality under pullback and also lacks a multiplicative structure.
There is a P^(1)\mathbb{P}^{1}-spectrum KGL_(S)inSH(S)\mathrm{KGL}_{S} \in \mathrm{SH}(S) that represents the so-called homotopy invariant KK-theory over an arbitrary base and is cartesian, so one could try s_(0)s_{0} KGL as a representing spectrum. Again, the problem is the functoriality of the slice filtration, but perhaps KGL would be easier to handle than the sphere spectrum in this regard.